Cummings left his London home shortly after he was diagnosed with the virus on March 30, and was later seen with his young son close to his parents' home in Durham, more than 250 miles (400 kilometres) away.
Under lockdown rules brought in on March 23, anyone with symptoms must self-isolate in their own homes. And people aged over 70 -- as Cummings' parents are -- are not allowed to receive visitors.
Britain's opposition Labour party said his actions suggested Cummings viewed himself as above the law and the smaller Liberal Democrats said it could be a resigning issue.
"The public have made extraordinary sacrifices during this pandemic and the lockdown. It cannot be one rule for those who set them and another for the British people," said Labour leader Keir Starmer.
An unnamed minister told the Daily Telegraph: "He's going to have to go. It's just arrogance."
But a spokesman for Prime Minister Boris Johnson said that Cummings had acted "in line with coronavirus guidelines."
"Owing to his wife being infected with suspected coronavirus and the high likelihood that he would himself become unwell, it was essential for Dominic Cummings to ensure his young child could be properly cared for," the spokesman said on Saturday.
"His sister and nieces had volunteered to help so he went to a house near to but separate from his extended family in case their help was needed," he added.
On March 24, deputy chief medical officer Jenny Harries had said that "although we are encouraging everybody to stay in their own households... clearly if you have adults who are unable to look after a small child, that is an exceptional circumstance."
Cummings also denied reports in the Guardian newspaper that police had spoken directly to him or his family over the issue.
Nevertheless, police in Durham, northeast England had confirmed on Friday that they received a tip-off on March 31 about someone who had travelled to the city from London.
"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 police said in a statement.
"In line with national policing guidance, officers explained to the family the guidelines around self-isolation and reiterated the appropriate advice around essential travel."
Government colleagues came to Cummings' defence, with foreign minister Dominic Raab tweeting that "two parents with coronavirus, were anxiously taking care of their young child.
"Those now seeking to politicise it should take a long hard look in the mirror," he added.
Cummings has been a highly divisive figure in British politics ever since masterminding the 2016 Brexit campaign alongside Johnson.
When Johnson was elected prime minister last year, he brought in Cummings as his top adviser.
Downing Street revealed at the end of March that Cummings was self-isolating with virus symptoms. Johnson was also infected, and ended up in intensive care.
Ed Davey, acting leader of the Liberal Democrats, said that if Cummings had broken the rules, "he will have to resign".
Earlier this month, leading epidemiologist Neil Ferguson quit a government panel on coronavirus after admitting breaking the lockdown rules by meeting his lover.
Scotland's chief medical officer, Catherine Calderwood, resigned in April after breaking her own rules to visit her second home twice.