The PM's decision to back his chief aide's lockdown trip to Durham has sparked fears that the government's coronavirus message will be undermined.
Some Tory backbenchers have called for Dominic Cummings to resign to ensure public confidence in future measures.
The row comes as plans to further ease lockdown restrictions will be discussed at a cabinet meeting later.
Education Secretary Gavin Williamson has said Mr Cummings "at no stage broke the law or broke the rules".
Speaking on BBC Radio 4's Today programme, Mr Williamson said that everyone, "whether they are anywhere in the country or working in No 10", was expected to abide by the law.
"That's what the prime minister expects, that's the assurance that he asked for, that is the assurance that he got."
Criticism of Boris Johnson's decision to take no action over Mr Cummings' 260-mile trip to his parents' home has come from all quarters.
Nineteen Tory MPs are calling for Mr Cummings to resign or be sacked, while others have joined Labour in calling for an inquiry.
Scotland's First Minister Nicola Sturgeon warned the consequences of Mr Johnson's decision could be "serious", and acting Lib Dem leader Sir Ed Davey said it undermined the prime minister's authority on the coronavirus crisis.
Senior Church of England bishops and scientists advising ministers on the pandemic have also strongly criticised the government's handling of the row.
Mr Johnson has defended Mr Cummings, saying he believed his senior aide had "no alternative" but to make the journey from London at the end of March for childcare "when both he and his wife were about to be incapacitated by coronavirus".
Durham's police chief, Steve White, has asked the force to "establish the facts concerning any potential breach of the law" surrounding Mr Cummings' visit to the county.
The prime minister is this week expected to set out details of plans to ease restrictions, which will reportedly include information about the reopening of some non-essential shops in June.
At Sunday's daily briefing, the prime minister confirmed the phased reopening of England's primary schools will begin on 1 June.
Boris Johnson failed to close down Cummings story
If Boris Johnson's decision to appear at Sunday's press conference was an attempt to close down the story about Dominic Cummings' behaviour during the lockdown by handling it himself, it failed completely.
It certainly was not an attempt to give the public the full information.
Instead, the prime minister refused to answer the questions that remain about the specifics of his adviser's visit or visits, to the north east of England while his team was telling the public again and again and again that they had to "stay at home".
Conservative backbencher David Warburton, MP for Somerton and Frome, said Mr Cummings was "damaging the government and the country that he's supposed to be serving".
Speaking on BBC Breakfast, Mr Warburton said: "We've not been offered the chance to interpret the rules, that's really not how it works otherwise there would be complete chaos."
The MP, who said his own father had died alone as a result of the coronavirus lockdown, said "people have made sacrifices" and "in those sacrifices there really hasn't been the choice to use instinct".
He added: "We've been tasked with following regulations laid down by the government."
Another Tory MP, Jason McCartney, said in a Facebook post that while it was important for people to show compassion during the crisis, Mr Cummings had to go because the "perceived hypocrisy of the rule makers potentially threatens the success of any future measures" under a second wave of coronavirus.
Theresa Villiers, a Tory MP and former cabinet minister, said she understood the row makes Mr Cummings' position "difficult", but she said: "I hope that he is able to stay because he's a highly effective adviser to the prime minister."
Meanwhile, some of the scientists that advise ministers were concerned the prime minister's decision to back Mr Cummings would undermine the message on controlling the virus, with one warning that "more people are going to die".
Stephen Reicher, a professor of social psychology who has advised the government on behavioural science during the pandemic, said the prime minister's backing of Mr Cummings made him feel "dismay".
He said trust was vital to maintaining public health measures. "You can't have trust if people have a sense of them and us," he told the BBC.
As one of those involved in SPI-B, the Government advisory group on behavioural science, I can say that in a few short minutes tonight, Boris Johnson has trashed all the advice we have given on how to build trust and secure adherence to the measures necessary to control COVID-19.— Stephen Reicher (@ReicherStephen) May 24, 2020
Speaking on ITV's Good Morning Britain, Prof Reicher - a member of the scientific pandemic influenza group on behaviours (SPI-B) - said research showed that the reason people observed lockdown was not for themselves but for the community.
"If you give the impression there's one rule for them and one rule for us you fatally undermine that sense of 'we're all in this together' and you undermine adherence to the forms of behaviour which have got us through this crisis," he said.
"The real issue here is that because of these actions, because of undermining trust in the government, because of undermining adherence to the rules that we all need to follow, people are going to die."
Two other members of SPI-B, which feeds into the Sage scientific advisory group, have backed Prof Reicher's comments - University College of London professors Robert West and Susan Michie.
It comes as the Bishop of Bristol, the Right Reverend Vivienne Faull, has accused the prime minister of having "no respect for people", while the Bishop of Leeds, the Right Reverend Nick Baines, said Mr Johnson was treating people "as mugs".
In other developments:
- More than 260,000 people have signed a petition calling for Dominic Cummings to be sacked since it was launched two days ago
- An investigation has been launched into a now-deleted tweet posted on the official UK Civil Service Twitter account, which asked "Can you imagine having to work with these truth twisters?"
- And the leader of the Liberal Democrats on Durham County Council tweeted that "a number of local residents have reported seeing Dominic Cummings on several occasions in April and May" and that she has referred the matter to Durham Police
Labour's shadow health secretary Jonathan Ashworth criticised the government in a series of tweets, saying that "in a pandemic everyone must adjust their behaviour to protect us all".
"How can Matt Hancock expect adherence to social distancing requirements while endorsing breaches like this," he said.
The shadow home secretary, Nick Thomas-Symonds, has written to Home Secretary Priti Patel about policing the lockdown, asking whether guidance on travel restrictions had been updated for parents with coronavirus symptoms.
Earlier, the former chief constable of Durham Police, Mark Barton, told BBC Breakfast that the prime minister's decision has "now made it exponentially tougher for all those on the frontline enforcing the lockdown".
The prime minister said he held "extensive" discussions on Sunday with Mr Cummings, who he said "followed the instincts of every father and every parent - and I do not mark him down for that".
One report alleges that a witness saw Mr Cummings in Barnard Castle, more than 25 miles from Durham, where he had been self-isolating, on 12 April.
Ministers to publicly support Mr Cummings include the Health Secretary Matt Hancock and Cabinet Office Minister Michael Gove.
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