The current national 17-day lockdown is due to be reviewed when it comes to an end on 9 November.
A senior Tory has said the party has to "admit we've misunderstood the mood of the country" over the extension free schools - adding the government will probably have to "think again" after voting against the move.
Sir Bernard Jenkin, Conservative chairman of the liaison committee, made the remarks as pressure builds on Boris Johnson to perform a U-turn on the issue.
Sir Bernard told the Sky News breakfast show Sophy Ridge on Sunday: "We have to admit we've misunderstood the mood of the country here... the public want to see the government taking a national lead on this, and I think the government will probably have to think again on that."
He added: "I think when you've got the chairman of the education select committee not supporting the government on this, and he's a Conservative, I think the government has to listen to the Conservative Party."
Sir Bernard was absent from the vote on free school meals but said he would have supported the government's position.
However, he added if there was another vote he would "wait to see what the government says and how they respond to the situation".
The Labour Party has threatened to bring the issue back to the House of Commons if Mr Johnson does not relent.
Anne Longfield, the children's commissioner, told Sophy Ridge on Sunday she thinks the government should extend the free school meals scheme.
She added: "I've been horrified by the debate, really disappointed about the debate over recent days.
"We're a wealthy country, it's 2020, and to have a debate to make sure hungry and vulnerable children should have enough to eat, is strikingly similar to what we would expect to see in chapters of Oliver Twist."
In relation to the possibility of the government performing a U-turn on the issue, Ms Longfield added: "I want this government to move from something that is a possibility and a discussion to something which is real."
It comes as more than 2,000 paediatricians have signed a letter telling the prime minister that childhood hunger should "transcend politics".
Brandon Lewis, the Northern Ireland secretary, defended the government's position on Sophy Ridge on Sunday.
He said: "Our policy on this is the right position because we are looking to deal with child poverty at the core."
Mr Lewis also said £63m was in place for local authorities to support people who are struggling.
Members of the Royal College of Paediatrics and Child Health said: "Few would disagree that one of our most basic human responsibilities is to ensure children have enough to eat.
"We call on the UK government to match the pledges of the Welsh and Scottish governments and the Northern Ireland Executive, to continue to provide children from low-income backgrounds with free meals over the coming weeks and to then extend this at least until the Easter school holiday, as they have done in Wales and Scotland."
The organisation also praised Marcus Rashford, 22, for his "powerful" work on the issue.
The Manchester United footballer, who was given free school meals when he was a child, has been campaigning on behalf of the 1.3 million children who are eligible for the scheme.
His petition, which calls for various measures to end child food poverty, is closing in on 800,000 signatures.
In June, he forced a government U-turn, which resulted in the meals being extended to cover the summer holidays.
But a motion calling for a further extension was rejected by Tory MPs last Wednesday.
Councils from both sides of the political spectrum joined businesses such as cafes and restaurants, many of them struggling financially due to the pandemic, in stepping up with offers of free meals for children in need.
They were encouraged by Rashford, who posted their efforts on his social media accounts.
Labour has said it will force another vote in the Commons if the government does not change its policy before the Christmas recess and extend free school meals during the school holidays until Easter 2021.
Shadow education secretary Kate Green called on Mr Johnson to meet Rashford's taskforce "as a matter of urgency" and even some of Mr Johnson's Conservative colleagues have said he should reconsider his stance.
Robert Halfon said meeting Rashford was a "no-brainer" and Tobias Ellwood said extending the meals over the holidays was a "simple and practical vehicle" to support families, adding that the government should "revisit" the issue.
Regarding the community effort to help feed vulnerable children, a Number 10 spokesman said: "I believe the PM said during PMQs that free school meals will continue during term time and that he wants to continue to support families throughout the crisis so they have cash available to feed kids if they need to."
Also, Sky's chief political correspondent Jon Craig revealed on Saturday that Rashford had written a "personal" letter to the prime minister at the start of September but the footballer did not receive a response.
The largest petition ever submitted to the Senedd has now passed 45,000 signatures.
The largest petition ever submitted to the Senedd has now passed 44,000 signatures.
Any change to self-isolation requirements would follow research by King's College London, which suggested just 10.9% of those traced as contacts of someone with Covid-19 remained at home for the full quarantine period.
The Welsh government has said it will review its ban on supermarkets selling non-essential items following a widespread backlash.
The news came just hours before a man was charged with criminal damage and contravention of coronavirus regulations after an incident at a Tesco store in Bangor.
Gwilym Owen, 28, from Anglesey, will appear before magistrates in Caernarfon on 24 November.
Video on social media from Friday showed a person at the store pulling plastic sheeting from shelves.
The Welsh government has been widely criticised for guidance saying certain sections of supermarkets must be "cordoned off or emptied, and closed to the public".
Affected items include electrical goods, telephones, clothes, toys and games, garden products and dedicated sections for homeware.
The reason put forward was to discourage people from spending too much time in shops and that it would be unfair if independent retailers were forced to close but supermarkets continued selling similar items.
Almost 43,000 people have signed a petition calling on politicians to reverse the ban, which it described as "disproportionate and cruel".
It said: "We do not agree that this is a prudent or rational measure, and will create more harm than good."
First Minister Mark Drakeford said the review would make sure "common sense is applied" to the rules governing the 17-day coronavirus lockdown.
Mr Drakeford said: "We'll be reviewing how the weekend has gone with the supermarkets and making sure that common sense is applied.
"Supermarkets can sell anything that can be sold in any other type of shop that isn't required to close. In the meantime, please only leave home if you need to."
Earlier, the Welsh government issued a statement saying the ban was "not for the sake of being difficult".
The leader of the Conservative opposition in the Welsh parliament, Paul Davies, branded the ban "madness" and said he has written to the presiding officer of the Senedd to call members back so they can debate the measures.
The popularity of the petition was a "clear sign" that people wanted the rule "scrapped immediately", he added.
The lockdown, which ends on 9 November, bans people from leaving their homes except for reasons such as buying food and medicine, providing care or taking exercise.
It also means people should work from home where possible.
Leisure, hospitality and tourism businesses are closed, as are libraries, community centres and recycling centres.
Places of worship are only allowed to open for funerals or weddings.
It has been estimated that the lockdown could cost the economy more than £500m.
On Saturday, Public Health Wales said 16 people had died with COVID-19, bringing the total to 1,772.
There were also 1,324 new cases of the virus confirmed, making a total of 41,577.
In addition a virus expert has warned figures have not “come tumbling down” despite Scotland’s miniature circuit-breaker across its central belt this month. For just under two weeks, Scotland has imposed stricter measures across certain areas including the closure of pubs and some other indoor venues.
The Government had suggested the measures would only be temporary, lasting for around 16 days.
However, the First Minister then announced the restrictions would be extended for a further week.
On Friday, she confirmed the country would take a “new approach” with a five-tier lockdown system.
Analysts say it could place businesses under restrictions for ‘the foreseeable future’ rather than relaxing them.
Emma McClarkin of the Scottish Beer & Pub Association told The Telegraph the situation “cannot continue”.
She added: “We were told that these measures were to be ‘short’ and ‘sharp’ but now the Scottish Government have gone back on that, leaving operators feeling betrayed.”
READ: Anti-lockdown protests descend into chaos as people refuse to wear masks - 10 arrested
He told the Sunday Telegraph: “If you close the pubs, you're going to stop pub outbreaks.
“You're addressing little bits of the transmission route, but not really getting at the fundamentals, which is making sure people who have the virus, and their contacts, are kept away from everybody else by self-isolating.”
In a press conference on Friday, Ms Sturgeon insisted there were “signs of progress” because of the lockdown restrictions.
She admitted cases were still rising but pointed to the rate of increase “which appears to be slowing down”.
First Minister also said it is “probably still too early for us to be seeing the full impact on case numbers of the restrictions on hospitality”.
Scotland’s new live-tier alert system will work similar to England’s one, in that it will apply regionally.
The alert levels go from tier 0 to tier 4, with tier 4 being referred to as ‘Lockdown’.
Under that level, non-essential shops will be required to close as they were in the UK-wide lockdown imposed in March, though schools will remain open and ‘some outdoor meetings’ will be permitted.
Level 0 allows for eight people from three households to meet indoors, and groups of up to 15 can meet outdoors.
In Scotland, 1,433 new cases of COVID-19 were reported on Friday with 11 new deaths.
