In a joint investigation, the Mirror and the Guardian reported that police intervened after receiving a tip-off that Mr Cummings had travelled to his family home in County Durham to self-isolate, despite Boris Johnson having declared a lockdown in England days beforehand.
According to the two papers, he was spotted a second time at the property on April 5.
Downing Street denied the police questioned Mr Cummings and he believes he behaved "reasonably and legally.”
A Number 10. spokesperson says it was “essential” for Mr Cummings to travel “to ensure his young child could be properly cared for”.
In a statement, Number 10 said the Prime Minister's aid had stayed in a separate house to his family.
But Labour have called on the Government provide more answers at the daily Downing Street Covid-19 press conference on Saturday.
In a statement on Saturday, a party spokesperson said: “We are still waiting for a clear explanation from Number 10 about Dominic Cummings’ actions.
“The public have made extraordinary sacrifices during this pandemic and the lockdown. It cannot be one rule for those who set them and another for the British people."
Sir Ed Davey, acting leader of the Liberal Democrats, has called for Mr Cummings to quit over the allegations.
“If Dominic Cummings has broken the lockdown guidelines he will have to resign, it is as simple as that,” the former energy secretary tweeted.
SNP Westminster leader Ian Blackford MP said the aide’s position was “completely untenable”, adding "he must resign or be sacked".
Mr Blackford said the Prime Minister had “serious questions” to answer regarding what he knew about Mr Cummings’ lockdown trip.
Speaking to BBC Radio Four, Mr Blackford branded the alleged actions the “height of irresponsibility for someone to think this is a reasonable course of action”.
“Here we have the highest official in Government, the closest confidant of the Prime Minister prepared to break the rules that the rest of us are being asked to obey," he said.
Even those in his own party - where he is a divisive figure - are said to have called for Mr Cummings to quit over the allegations. ITV News political correspondent Paul Brand said a backbencher had told him he had his "fingers crossed" for the prime minister's aide's resignation.
"I think it’s fair to say that Conservative MPs won’t be riding to the rescue,” they told him.
Former Conservative MP Anna Soubry wrote on Twitter: "All I can think of is all those people who kept to the rules & didn’t say farewell or hold the hand of a dearly loved family member before they died nor attend their funeral. Govt has been consistent - do as we say not as we do. Arrogant & elitist.
But friends of Mr Cummings suggested he would be going nowhere saying he was "not remotely bothered by the story."
In a statement, No10 spokesperson said: “Owing to his wife being infected with suspected coronavirus and the high likelihood that he would himself become unwell, it was essential for Dominic Cummings to ensure his young child could be properly cared for.
“His sister and nieces had volunteered to help so he went to a house near to but separate from his extended family in case their help was needed.
“His sister shopped for the family and left everything outside. At no stage was he or his family spoken to by the police about this matter, as is being reported. His actions were in line with coronavirus guidelines. Mr Cummings believes he behaved reasonably and legally.”
Mr Cummings is alleged 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 in the city after reports a person had travelled there from London.
Cabinet ministers have previously supported the decision of those involved in the Government’s response to Covid-19 resigning after disobeying the lockdown.
, the epidemiologist whose modelling with Imperial College London prompted the lockdown, quit as a member of the Scientific Advisory Group for Emergencies (Sage), the body advising ministers during the crisis, after it was found he was visited by his married lover.
Mr Cummings, formerly one of the key strategists behind the Vote Leave campaign, wrote about his experience of isolating with his wife, Mary Wakefield, in the Spectator magazine.
He wrote that “at the end of March and for the first two weeks of April I was ill, so we were both shut in together”.
Ms Wakefield, also writing in the Spectator, which she is an editor for, said she became ill with Covid-19 symptoms before her husband did.
“I felt breathless, sometimes achy, but Dom couldn’t get out of bed,” she said.
“Day in, day out for ten days he lay doggo with a high fever and spasms that made the muscles lump and twitch in his legs. He could breathe, but only in a limited, shallow way.”
No 10 has been approached for comment.
Downing Street had regularly refused to confirm where Mr Cummings was self-isolating after news broke that he was ill, possibly with coronavirus.
Mr Cummings has since returned to work at No 10.