Cummings on Saturday told a throng of reporters while leaving his house that it was "a question of doing the right thing, it's not about what you guys think", while rebuking the grounp for not obeying social distancing rules.
But according to reports in the Observer and Sunday Mirror, the former Vote Leave campaign co-ordinator made a second trip to Durham and was seen there on 19 April - five days after being photographed on his return to Westminster.
But Durham Police said officers spoke to Mr Cummings' father over the phone on April 1.
But the trip to Barnard Castle on April 12, if correct, would call that testimony into question. "I do not mark him down for that".
Government advice had been for people to stay at home during the first weeks of lockdown. "It couldn't have been clearer".
The newspapers have since reported that Cummings was seen in northern England on other occasions.
He reportedly was heard commenting on how "lovely" the bluebells were during an early-morning Sunday stroll with his journalist wife Mary Wakefield.
Fellow Conservative MP Simon Hoare has called for Cummings to "consider his position" and Tory MP Damian Collins has said the government "would be better without him".
Culture Secretary Oliver Dowden tweeted shortly after the daily press conference had finished, saying: "Dom Cummings followed the guidelines and looked after his family".
However, the main opposition Labour party said the reports suggested Cummings viewed himself as above the law, while the smaller Liberal Democrats party has hinted that the incident could mean Cummings should resign.
After the latest revelations emerged, a Labour source said: "If these latest revelations are true, why on earth were Cabinet ministers sent out this afternoon to defend Dominic Cummings?".
A petition calling for Mr Cummings to be sacked has also been circulated and has so far reached almost 75,000 signatures.
He said Johnson must explain why "it looks like there is one rule for those at the center of government and one rule for rest of the country".
Responding to the allegations, a No 10 spokeswoman said: "Yesterday the Mirror and Guardian wrote inaccurate stories about Mr Cummings".
Peter Aldous, MP for Waveney said today he could understand the senior adviser's decision to make the first trip from London to his parents' home in Durham County, but said he would reserve judgement on claims surrounding a second journey until the facts were established.
Downing Street had earlier said that it would "not waste time" replying to the new allegations from "campaigning newspapers".
The changes marked a stark escalation of the government's response to the pandemic.
"He and his wife had symptoms, if not indeed had Covid-19, and were concerned for the welfare of their child".
"Durham Constabulary deemed that no further action was required".
Number 10 Downing Street special advisor Dominic Cummings gets out of his auto as he arrives at his home in London on May 23, 2020 after allegations he broke coronavirus lockdown rules by travelling across the country in March.
In a televised address to the nation on March 23, Prime Minister Boris Johnson announced unprecedented limits on where and how people can meet and gather during the continuing coronavirus.
Government ministers defended him as a father concerned for the welfare of his child who travelled to be near his family but self-isolated in a separate building, away from his elderly parents.