Mr Cummings made the alleged journey despite the official government advice warning against all but essential travel.
A member of the public is said to have spotted Mr Cummings and the sighting was reported to 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.”
ITV News Political Editor Robert Peston explains how Mr Cummings' alleged movements 'look extraordinary' as he attends the SAGE meetings
Mr Cummings tested positive for coronavirus, having developed symptoms over the weekend of 28 and 29 March.
There was a further alleged sighting of Mr Cummings on 5 April with a young child near his parents' home in the area.
Downing Street has always refused to confirm where Mr Cummings spent lockdown.
The Labour Party has demanded a “very swift explanation” from Downing Street over the reports.
A Labour spokesman said: “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.
“The British people do not expect there to be one rule for them and another rule for Dominic Cummings. Number 10 needs to provide a very swift explanation for his actions.”
Mr Cummings is the latest person in the public eye to face criticism for allegedly breaking lockdown rules.
Sir Ed Davey, acting leader of the Liberal Democrats, has called for Mr Cummings to walk 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”.
“He must resign or be sacked,” he added.
Scotland’s chief medical officer, Catherine Calderwood, resigned after travelling to a holiday home, while Professor Neil Ferguson quit SAGE after it was reported his girlfriend visited him during lockdown.