Dominic Cummings travelled hundreds of miles from London to County Durham during the lockdown when he had virus symptoms, the BBC has been told.
A source close to the PM’s chief aide confirmed reports he and his wife went to his parents’ home to self-isolate.
But the BBC’s Laura Kuenssberg said she had been told Mr Cummings did not break government guidance because he and his wife stayed in a separate building.
The story was first reported by the Mirror and Guardian newspapers.
Labour said No 10 needed to provide a “very swift explanation” for Mr Cummings’ actions.
“If accurate, the prime minister’s chief adviser appears to have breached the lockdown rules. The government’s guidance was very clear: stay at home and no non-essential travel,” a spokesman said.
“The British people do not expect there to be one rule for them and another rule for Dominic Cummings.”
The Scottish National Party’s Westminster leader Ian Blackford said Mr Cummings should resign or be dismissed by Boris Johnson.
“This is a key test of leadership for Boris Johnson. People must have confidence that the Tory government is following its own rules – not being investigated by the police for breaking them,” he said.
And Ed Davey, acting leader of the Liberal Democrats, added: “If Dominic Cummings has broken the guidelines he will have to resign, it is as simple as that.”
The Mirror and the Guardian reported Mr Cummings, the former Vote Leave chief who was the architect of the PM’s Brexit strategy, had been approached by the police.
Durham Constabulary said: “On Tuesday, March 31, our officers were made aware of reports that an individual had travelled from London to Durham and was present at an address in the city.
“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 guidelines around self-isolation and reiterated the appropriate advice around essential travel.”
Mr Johnson’s positive test for coronavirus was announced on 27 March, and Downing Street said at the time that Mr Cummings did not have symptoms. On 9 April, the PM’s official spokesman said Mr Cummings was self-isolating with symptoms of the virus and was not back in No 10.
Mr Cummings was next photographed at Downing Street on 14 April.
A source close to Mr Cummings said he had not been spoken to by police, and that he had made the trip because his parents could help care for his young child while he and his wife were both ill with symptoms of coronavirus.
No 10 declined to comment.