Police have confirmed they attended a property in County Durham after it emerged that Dominic Cummings, who is Boris Johnson's chief adviser, travelled more than 260 miles from his London home during the lockdown.
It is suggested he stayed with relatives while he and members of his immediate family were suffering from coronavirus-related symptoms.
The SNP's Westminster leader, Ian Blackford, said there are "serious questions" for Mr Johnson to answer over reports that members of his inner circle knew Mr Cummings left London only days after the PM had issued strict instructions for people to stay at home and not to drive long distances - or else face punishment.
According to a joint investigation by the Guardian and Daily Mirror, Mr Cummings was spotted twice in the North East over the course of almost a week, between March 31 and April 5 - eight days after lockdown began.
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Speaking to BBC Radio 4's Today programme, Mr Blackford said: "What I find interesting is that (according to reports) members of Downing Street knew about this so, first and foremost, Boris Johnson has serious questions to answer over what now appears to be a cover-up.
"The Prime Minister must explain exactly when he knew about the breaking of the rules, whether he sanctioned it, why Cummings wasn't sacked immediately and why it appears that he tried to cover it up, not telling the public until the newspaper(s) broke the story, eight weeks later, last night."
He branded the alleged actions the "height of irresponsibility", and added: "Demonstrably, this is an individual who has broken the advice he has been, in many cases, the architect of delivering."
The Labour Party said the "country deserves answers" at the daily Downing Street press conference on Saturday.
Dave Penman, general secretary of the FDA, the union that represents senior civil servants, said Mr Johnson has a duty, as set out in the ministerial code, to explain his aide's actions.
He told the Today programme: "The Prime Minister's silence is simply not good enough.
"If he doesn't think Dominic Cummings has flouted the rules, when there seems to be such an obvious case to answer, then he needs to explain that to maintain the integrity of his position.
"He has known about this for six weeks."
But, despite growing calls for Mr Cummings to quit, friends of the mastermind behind the Vote Leave campaign suggested he would be going nowhere.
One told the PA news agency: "He isn't remotely bothered by this story, it's more fake news from the Guardian.
"There is zero chance of him resigning."
Mr Cummings is said to have been present at his family home when police from Durham Constabulary turned up on March 31, following a call from someone reporting they had seen Mr Cummings in the area.
Durham Police confirmed officers had spoken to the owners of an address on after reports that a person had travelled there from London.
A spokesman said: "Officers made contact with the owners of that address, who confirmed that the individual in question was present and was self-isolating in part of the house.
"In line with national policing guidance, officers explained to the family the arrangements around self-isolation guidelines and reiterated the appropriate advice around essential travel."
Downing Street had previously confirmed that Mr Cummings had started displaying coronavirus symptoms "over the weekend" of March 28 and 29.
The same day, as police spoke with members of Mr Cummings's family, his boss, Mr Johnson, was admitted to hospital with coronavirus symptoms, and later required treatment in intensive care.
An unnamed neighbour told the two papers that Mr Cummings was seen in the garden five days after the police visit as Abba's Dancing Queen was playing loudly.
"I got the shock of my life, as I looked over to the gates and saw him," they said.
"I recognised Dominic Cummings - he's a very distinctive figure."
Downing Street and Mr Cummings are yet to comment on the allegations.