The prime minister's chief aide Dominic Cummings is facing fresh allegations that he breached lockdown rules.
He and the government had said he acted "reasonably and legally" by driving from London to County Durham while his wife had coronavirus symptoms.
No 10 says this is "inaccurate" but some Tory MPs have called for Mr Cummings to consider his position.
Labour has called for an urgent inquiry into the allegations, while government ministers rallied around Mr Cummings on Saturday and defended his conduct.
Matt Hancock and Michael Gove were among those to back Mr Cummings for self-isolating at a property adjacent to other family members in case he and his wife needed help with childcare.
Mr Cummings told reporters outside his home on Saturday that he would not be resigning and had done the "right thing" by travelling 260 miles with his wife and young son to be near relatives when she developed Covid-19 symptoms at the end of March.
The two newspapers have now reported witnesses saw Mr Cummings in Barnard Castle, more than 25 miles from Durham, on 12 April.
On 14 April, he was seen in London. According to a witness, he was spotted again near Durham in Houghall Woods on 19 April.
Mr Cummings is yet to publicly respond to the new claims, but the Sunday Telegraph reports he told Downing Street he left Durham on 13 April, and that the claim he made a second trip from London was "totally false".
Transport Secretary Grant Shapps told the BBC's Andrew Marr that the claims of a second trip were "untrue".
When asked if Mr Cummings was going to resign, Mr Shapps replied: "No."
But a growing number of backbench Tory MPs have now called on Mr Cummings to resign.
'Position is untenable'
Ex-chairman of the European Research Group (ERG) Steve Baker told the BBC: "The country can't afford this nonsense, this pantomime, Dominic should go and we should move on and deal with things that matter in people's lives."
Tory Sir Roger Gale said there "cannot be one law for the prime minister's staff and another for everyone else".
While as a father and as a grandfather I fully appreciate Mr Cummings’ desire to protect his child. There cannot be one law for the Prime Minister’s staff and another for everyone else. He has sent out completely the wrong message and his position is no longer tenable.— Sir Roger Gale MP (@SirRogerGale) May 24, 2020
Conservative MP Caroline Nokes tweeted: "There cannot be one rule for most of us and wriggle room for others."
Colleague Simon Hoare has called for Mr Cummings to "consider his position", Tory MP Damian Collins has said the government "would be better without him" and MP Craig Whittaker has said Mr Cummings' position "is untenable".
Labour's shadow policing minister Sarah Jones told Sky's Sophy Ridge on Sunday that people "are feeling rightly angry".
"I think people are rightly feeling is it one rule for us and one rule for people at the top," she said.
In response to the fresh claims, Downing Street said: "Yesterday the Mirror and Guardian wrote inaccurate stories about Mr Cummings.
"Today they are writing more inaccurate stories including claims that Mr Cummings returned to Durham after returning to work in Downing Street on 14 April.
"We will not waste our time answering a stream of false allegations about Mr Cummings from campaigning newspapers."
Downing Street has also denied that police spoke with family members of Mr Cummings "about this matter".
By Helen Catt, political correspondent
After an apparently co-ordinated show of support from some of the most senior Conservatives yesterday, today the first cracks may be starting to show.
Steve Baker has become the first Tory MP to break ranks and call for Dominic Cummings to go.
As one of Parliament's most prominent Brexiteers, his intervention is significant.
The tone of his criticism even more so - accusing Mr Cummings of regarding "accountability with contempt".
His view is that Boris Johnson is expending too much political capital on trying to save his adviser.
The next few hours will see how many more follow suit.
If there's enough backbench unrest, it will leave Boris Johnson with an unappealing choice to make: oust a highly-valued adviser or risk upsetting the party to keep him.
Other opposition parties have also renewed their calls for the prime minister's adviser to go.
The SNP's Ian Blackford said Mr Cummings "has to leave office", while acting Liberal Democrat leader Sir Ed Davey told BBC Radio 5 Live: "If Dominic Cummings has not been sacked by tomorrow, I think the prime minister's judgement is in serious doubt."
Following the fresh reports concerning the alleged second visit to County Durham, a Labour source said: "If these latest revelations are true, why on earth were cabinet ministers sent out this afternoon to defend Dominic Cummings?"
Government advice had been for people to stay at home during the first weeks of lockdown. Self-isolation at home continues to be advised for those with coronavirus symptoms.
England's deputy chief medical officer Dr Jenny Harries said all health guidance should be applied with "common sense".
It comes as the government announced 282 more people had died with coronavirus since Friday, across all settings, bringing the total to 36,675.
When asked by reporters outside his home on Saturday whether his travelling to Durham looked good, Mr Cummings said: "Who cares about good looks? It's a question of doing the right thing. It's not about what you guys think."
Asked whether he would reconsider his position, he said: "Obviously not."
Mr Cummings masterminded the 2016 Vote Leave campaign before being made Prime Minister Boris Johnson's chief political adviser.
In other developments:
- Rising costs from the coronavirus pandemic could see care homes in England "go to the wall", sector leaders have warned
- People arriving in France from the UK from 8 June will have to quarantine for 14 days
- The government has pledged to give £283m towards improving safety and restoring services on bus and light rail services during the pandemic