pressure to say whether he attended a drinks event in the Downing Street garden during the first lockdown.
Witnesses have said the PM and his wife were among 30 people at the event on the 20 May 2020.
Senior Conservatives have joined Labour in saying he must explain his actions.
Labour’s deputy leader Angela Rayner said if he was there, and he lied about it, his position is “untenable.”
The prime minister has so far declined to say whether or not he attended the event.
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On Monday, ITV published a leaked email from Mr Johnson’s principal private secretary, Martin Reynolds, inviting staff to “socially distanced drinks in the No 10 garden” on 20 May 2020 – when lockdown rules in England banned large outdoor gatherings.
The invitation encouraged people to “bring your own booze” and “make the most of the lovely weather”.
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Labour’s deputy leader Angela Rayner told BBC Breakfast if Mr Johnson had broken Covid rules and lied to the public and parliament, then his position as prime minister was now “untenable”.
“The prime minister cannot break the law, lie to the British public and not be held to account for that,” she added.
Liberal Democrat leader Sir Ed Davey told BBC Breakfast if the prime minister had a “shred of decency left”, he should resign today.
He added Mr Johnson was “incapable of leading our country through this public health crisis”, accusing him of being a “threat to the health of the nation” as the public would not listen to anything he had to say.
If the prime minister won’t resign, it was the duty of Conservative MPs to “do the right thing” and ensure he does “in the interests of the country”, Sir Ed argued.
Mr Johnson has said it is a matter for Sue Gray – the senior civil servant investigating a series of reported parties in Downing Street and in Whitehall in 2020 – to determine what happened.
The PM’s official spokesman has also refused to confirm whether Mr Johnson attended the May 2020 Downing Street gathering. He added that the prime minister retained full confidence in Mr Reynolds.
But backbencher Nigel Mills told BBC Newsnight an inquiry was not needed to establish whether the prime minister was there.
“He knows whether he was there or not. Just come out and say what happened,” he said.
Mr Mills said it would be “utterly untenable” for any senior figure who attended the gathering to be responsible for setting Covid-19 policy.
“If the prime minister knowingly attended a party, I can’t see how he can survive,” he said.
Conservative MP Huw Merriman insisted “more clarity” was needed from the prime minister on what took place and whether he attended, adding that people were “in the dark” about what went on.
He told BBC Radio 4’s Today programme that he didn’t believe Mr Johnson should resign, adding that people should be judged on the positives and negatives, citing government achievements including the successful vaccination rollout and the prime minister “holding his nerve” on refusing to introduce further Covid rules in England.
Scottish Conservative leader Douglas Ross said Mr Johnson must quit if he was found to have broken Covid rules.
Sir Charles Walker, vice-chairman of the Conservative backbench 1922 Committee, said there was a lot of anger over what had happened, adding that the prime minister urgently needed to rebuild public trust.