The calls came after the prime minister staked his authority on a full-throated defence of his most senior adviser, saying the public would understand why he drove 260 miles while they were being told to stay at home.
Addressing the controversy at the daily Downing Street press conference, Mr Johnson said Mr Cummings had acted “with integrity” and had to travel to seek childcare for his young son.
The prime minister had earlier come under mounting pressure to sack Mr Cummings after a series of Tory MPs broke ranks to suggest he had to go.
However, Mr Johnson made it clear that Mr Cummings would remain in his post, saying he had acted legally and responsibly and “followed the instincts of every father and every parent”.
But he skirted over questions about allegations Mr Cummings took a 30-mile day trip from his family home at the height of the coronavirus outbreak.
The day Mr Cummings was spotted in Durham was the day his uncle, Lord Justice Laws, died, his parents told the New Statesman. The former senior judge had Covid-19 when he died.
Questions also persist about whether or not Mr Cummings stopped during the journey from London to Durham.
Labour, the Liberal Democrats and the SNP have already called for Sir Mark Sedwill, the cabinet secretary, to launch an official inquiry.
One MP from the so-called “blue wall” of seats in the north of England, which switched to the Conservatives during last December’s general election, told The Independent: “We are absolutely f*****g livid. Our feelings have been made widely known, to the whips and to others. They have to launch an inquiry into what exactly the hell happened. We need to know the facts. And it needs to be done by Sedwill or someone like that. Somebody needs to get to the bottom of this and quickly. An inquiry was good enough when it came to Priti [Patel], after all.
“What none of us can understand is why facts are still in dispute. Whether it is the trip to [Barnard Castle] or broader questions of why it is good enough for our constituents, isolating at home, but is not good enough for Dominic Cummings.”
On Sunday night No 10 declined to comment on calls by Tory MPs for an inquiry. Priti Patel, the home secretary, is currently the subject of an inquiry into her conduct following allegations of bullying.
In an apparent effort to distract from the Cummings controversy, Mr Johnson also used the press conference to announce his intention for early-years centres to reopen and primary schools to restart lessons for pupils in reception class, year one and year six from 1 June in England.
The decision must be formally confirmed at the nine-week review of lockdown due on Thursday, but the prime minister said he wanted to give teachers and parents time to prepare.
Unusually, journalists at Sunday’s briefing were given only a single question with no chance to challenge the PM’s answers.
Ministers had expressed hopes that Mr Johnson’s defence of Mr Cummings would draw a line under the row, but after the press briefing yet another Tory, Paul Maynard, the MP for Blackpool North and Cleveleys, tweeted to say Mr Cummings’s position was “untenable”.
Tory grandee Sir Malcolm Rifkind also said he was saddened that Mr Cummings was not “big enough” to do his “duty as an official” and resign even if he believed he was innocent.
Tory MPs further complained they were facing a flood of angry emails from constituents, a significant proportion of them Conservative Party members, over the controversy.
One former Conservative cabinet minister said that Mr Johnson had “screwed MPs by tying this to himself. Now you are either for him or against him”.
There were more woes for the prime minister late on Sunday as Mr Cummings was reported to police for an alleged breach of lockdown and an ex-chief constable accused him of risking lives by “selfishly” ignoring the rules, according to The Guardian and The Daily Mirror, which jointly broke the original story. An eyewitness told The Observer and The Sunday Mirror he had seen Mr Cummings in Barnard Castle on 12 April, at the height of the pandemic.
Sir Keir Starmer, the Labour leader, said Mr Johnson’s defence of Mr Cummings was an “insult” to the sacrifices made by the British people during lockdown, and added: “If I was prime minister I would have sacked him.”
In an appearance on Sky News earlier in the day, transport secretary Grant Shapps said he did not know if Mr Cummings had taken the day trip.
He told Sophy Ridge on Sunday: “I certainly know that the first [allegation] you mention, of travelling back up, I know that is not true. I’m afraid I don’t know [about Barnard Castle] but if that date was true that would have been outside the 14-day period. But I’m afraid I don’t have the information on that.”
Professor Neil Ferguson, the epidemiologist whose modelling prompted the lockdown, quit as a government adviser for flouting the rules when he was visited at his home by his alleged lover.