Downing Street confirmed that Cummings was infected and self-isolating with virus symptoms, but did not disclose any trips.
The source insisted Mr Cummings did not break official guidance because the couple stayed in a separate building.
Police in Durham confirmed that on March 31 they had been tipped off about someone who had travelled to the city from London and did a routine check. A spokesman for the Labour Party told the BBC: "The British people do not expect there to be one rule for them and another rule for Dominic Cummings".
Cummings was later spotted near his parents' home with a small child, believed to be his son, the report 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".
Downing Street has so far refused to comment.
"If precise, the head of state's principal advisor shows up to have actually breached the lockdown rules", a Labour spokesperson said.
The 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. Downing Street said he was self-isolating at home.
"Officers explained to the family the guidelines around self-isolation and reiterated the appropriate advice around essential travel".
"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".
Scotland's Chief Medical Officer Catherine Calderwood resigned on April 5 after she broke her own advice to stay at home by visiting her second home on two weekends.
The Scottish National Party's parliamentary leader, Ian Blackford, said Mr Johnson should fire Mr Cummings.
These were the details of the news Coronavirus: Boris Johnson's top aide broke lockdown rules for this day.
At the time, Johnson's government was advising people to only go out for essential purposes to slow the spread of coronavirus, and for anyone with symptoms to stay at home.
Alastair Campbell hit out at the BBC Political Editor. In 1994, Cummings graduated from Exeter College, Oxford, with a degree in Ancient and Modern History.
In recent years, Cummings has worked as Chief of Staff to then-Education Secretary Michael Gove, before masterminding the successful Vote Leave Brexit campaign in 2016.
Pressure has grown on Prime Minister Boris Johnson to sack his closest aide after the allegations emerged.
As of 9am on Friday 22 May, there are 1,898 confirmed cases of COVID-19 in County Durham, out of a population of around 530,000.