The PM's chief aide Dominic Cummings is facing calls to resign after it emerged he travelled from London to his parents' home in Durham with coronavirus symptoms during lockdown.
Mr Cummings and his wife, who was also unwell, stayed at his parents' home while self-isolating.
Labour demanded No 10 provided a "swift explanation" for Mr Cummings' actions.
A source close to Mr Cummings denied a breach of the coronavirus rules, saying the couple needed childcare help.
They told the BBC's political editor Laura Kuenssberg that the couple had stayed in a separate building at the property.
The government has ordered anyone with coronavirus symptoms to self-isolate at home and not leave - even for essential supplies - for seven days.
Meanwhile, Prime Minister Boris Johnson said in March that children should not be left with older grandparents or older relatives "who may be particularly vulnerable or fall into some of the vulnerable groups".
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."
Mr Cummings has attended meetings of the Scientific Advisory Group for Emergencies (Sage) - the scientific body which gives independent advice that shapes the government's coronavirus response.
The Scottish National Party's Westminster leader Ian Blackford said Mr Cumming's position was "untenable" and that the prime minister has "serious questions" to answer about the reports.
He told BBC Radio 4's Today programme there was "nothing in the guidance that justifies travelling more than 250 miles".
"Dominic Cummings has to do the right thing, and if he doesn't resign, Boris Johnson should sack him and he should do that this morning," Mr Blackford said.
"When you have a situation that at the highest level of government that [lockdown] rules aren't being followed then I think people expect action to be taken."
Meanwhile, Dave Penman, general secretary of the FDA, the union that represents senior civil servants, said the prime minister was responsible for the actions of his chief aide, and called for Mr Johnson to explain reports that Mr Cummings broke lockdown rules.
He told BBC Radio 4's Today programme that "essentially, when he [Mr Cummings] says or does something he is doing it in the prime minister's name".
It comes after other high-profile figures involved in tackling the pandemic have resigned for breaching lockdown restrictions, including Scotland's former Chief Medical Officer Catherine Calderwood and leading scientist Prof Neil Ferguson.
Dr Calderwood resigned in April after making two trips to her second home during the coronavirus lockdown.
While Prof Ferguson quit as a government adviser on coronavirus after it was reported that a woman he was said to be in a relationship with visited his home in lockdown.
By Leila Nathoo, political correspondent
At the time Dominic Cummings had coronavirus symptoms, there was only a limited set of reasons for which people were allowed to leave their homes.
And the advice for anyone was - and is - not to leave home at all for at least seven days.
Remember there have already been other senior figures involved in tackling the pandemic who have had to resign for breaching lockdown restrictions.
A source close to Dominic Cummings is insistent that he didn't break the rules.
But for those at the top to be perceived to even be stretching the rules is damaging.
Along with triggering accusations of hypocrisy, it risks prompting people to question why they should be following the rules, if those involved in imposing them are not.
On 30 March, it was reported that Mr Cummings, the former Vote Leave chief who was the architect of the PM's Brexit strategy, had developed symptoms of the coronavirus and was self-isolating at home.
Durham Police confirmed officers had spoken to the owners of an address after its officers were made aware on 31 March of reports a person had travelled there from London.
A spokesman said that owners of the address confirmed the individual "was present and self-isolating in part of the house", and officers "explained to the family the guidelines around self-isolation and reiterated the appropriate advice around essential travel".
The source close to Mr Cummings denied that police spoke to him.
Ed Davey, acting leader of the Liberal Democrats, said: "If Dominic Cummings has broken the guidelines he will have to resign, it is as simple as that."
And speaking on BBC Newsnight, former Conservative MP David Liddington, who was de facto deputy PM under Theresa May, warned that the "readiness of members of the public to follow government guidance more generally is going to be affected by this sort of story".
Government advice on 31 March was for the public to stay at home and only leave for clearly defined reasons, including to exercise, essential shopping or for medical needs.
At the time - and as remains the case - those with symptoms of coronavirus were told to self-isolate at home and not leave even for essential supplies, if possible, for seven days.
It was and also remains not permitted to leave your house to visit friends and family in their home, while government advice for those aged 70 and over continues to be that they should minimise contact with others outside their household.
'Fever and spasms'
Mr Cummings' wife, Mary Wakefield, wrote about their experience of the disease in the Spectator magazine.
"I felt breathless, sometimes achy, but Dom couldn't get out of bed," she said. "Day in, day out for 10 days he lay doggo with a high fever and spasms that made the muscles lump and twitch in his legs."
Also in the magazine, Mr Cummings wrote about his experience of self-isolating with his wife, saying that "at the end of March and for the first two weeks of April I was ill, so we were both shut in together".
At the time Mr Johnson's positive test result for coronavirus was announced on 27 March, Downing Street said that Mr Cummings was not unwell. But within days it was confirmed that the strategist had started displaying coronavirus symptoms.
Mr Cummings was next photographed at Downing Street on 14 April after his recovery.
In other developments:
- Police are "wary" on patrol amid a rise in spitting attacks during the coronavirus pandemic
- People arriving in the UK must self-isolate for 14 days from 8 June to help slow the spread of coronavirus, the government has said
- The founder of the UK's weekly applause for front-line workers has said it should end next Thursday
- A care home in Oxfordshire has told how it locked down early to protect its residents
- And the BBC's medical correspondent Fergus Walsh has tested positive for coronavirus antibodies