"It can not be one rule for those who set them and another for the British people", Labour leader Keir Starmer said.
Johnson's office said in a statement that Cummings made the trip because his wife was showing coronavirus symptoms, he correctly thought he was likely to also get sick, and relatives had offered to help look after the couple's toddler son.
When pressed by reporters on Saturday as to how his actions looked to the public, Cummings responded angrily, asking the Daily Mail "who cares about good looks".
So far Boris Johnson has resisted calls to sack him and Cummings himself has refused to resign.
Durham Police say officers went to a house on March 31 and "explained to the family the guidelines around self-isolation and reiterated the appropriate advice around essential travel". Johnson's most senior ministers, including Michael Gove, foreign secretary Dominic Raab and health secretary Matt Hancock, defended Cummings.
British media reports previously said the idea of pushing for "herd immunity", which was one of the official paths discussed the United Kingdom would take before the virus hit the country, came from Dominic Cummings.
The reports could put Mr Cummings in a hard position, after several senior people involved in the country's response to the coronavirus outbreak were forced to resign when it emerged that they had broken the lockdown rules.
The spokesperson added: "Mr Cummings believes he behaved reasonably and legally". "It's not about what you guys think." said Cummings, who also berated the journalists for failing to keep 2 metres (6 1/2 feet) apart in line with social distancing rules.
"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", the force said in a statement.
Acting Lib Dem leader Sir Ed Davey said that the issue risked distracting the government from the pandemic while the SNP's Ian Blackford said Mr Cummings "has to leave office".
Dorset's police and crime commissioner, Martyn Underhill, mentioned such a high-profile breach of the foundations by Cummings would make it tougher for police to implement the lockdown.
While Cummings is known for his defiance of conventional wisdom as well as his propensity to upset the established order, the reports could put Cummings in a hard position, as several other senior officials who had reportedly broken the lockdown rules were forced to resign from their posts.
Literally rewriting the lockdown rules before our eyes just to save the Prime Minister's chief adviser. "Number 10 needs to provide a very swift explanation for his actions".
The two newspapers later reported that Cummings was spotted again in the Durham area on April 19, after he had recovered from the virus and returned to work in London.
Days before, Prime Minister Johnson had tested positive for the virus, and he was self-isolating in Number 10 Downing Street.
Dominic Cummings may have some explaining to do.
Prof Neil Ferguson, 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. Scotland's chief medical officer, Catherine Calderwood, stepped down after she was caught making two trips to her second home.