Downing Street says Boris Johnson accepts Matt Hancock’s apology for breaking social distancing rules and “considers the matter closed”.
The health secretary said he has “let people down” and is “very sorry” after pictures emerged of him kissing an aide.
Questions about Mr Hancock hiring former lobbyist Gina Coladangelo surfaced after pictures in The Sun newspaper showed the health secretary, who is married, embracing the aide in his office.
Facing questions from reporters at a regular Westminster briefing, the prime minister’s spokesman did not say whether Mr Johnson had asked the health secretary to resign, nor if Hancock had offered the PM his resignation.
The spokesman also refused to address suggestions that Mr Hancock had broken the law, repeatedly pointing reporters to the health secretary’s statement.
According to the newspaper, the images, which appear to be from CCTV footage, were taken on 6 May from the Department of Health and Social Care building.
At the time the picture was reportedly taken, hugging anyone outside your household or bubble was not allowed.
The government’s roadmap out of COVID restrictions said people should maintain their distance from anyone not in their household or support bubble until 17 May.
Labour and the Liberal Democrats have called on the PM to sack the health secretary, but Mr Hancock made clear in his statement that he wants to continue.
“I remain focused on working to get the country out of this pandemic, and would be grateful for privacy for my family on this personal matter,” he said.
Mr Hancock’s statement comes after he did not appear at a planned constituency event on Friday morning at Newmarket Racecourse.
The Sunday Times reported in November that Mr Hancock had failed to declare he had appointed Ms Coladangelo as an unpaid adviser on a six-month contract last March and later gave her a role on the board of the Department of Health.
Ms Coladangelo, who is listed on the department’s website as a non-executive director, is the marketing and communications director at British retailer Oliver Bonas, which was founded by her husband Oliver Tress.
Her LinkedIn profile says she has been working as a non-executive director at the department since September 2020 and was at Oxford University at the same time as the health secretary.
Mr Hancock has been married to his wife Martha for 15 years and they have three children together.
The health secretary’s wife did not comment as she returned to the couple’s home in northwest London on Friday afternoon.
In a statement before Mr Hancock’s comments were released, Labour Party chair Anneliese Dodds said: “If Matt Hancock has been secretly having a relationship with an adviser in his office – who he personally appointed to a taxpayer-funded role – it is a blatant abuse of power and a clear conflict of interest.
“The charge sheet against Matt Hancock includes wasting taxpayers’ money, leaving care homes exposed and now being accused of breaking his own COVID rules.
“His position is hopelessly untenable. Boris Johnson should sack him.”
Reacting to his statement, Ms Dodds tweeted: “He set the rules. He admits he broke them. He has to go. If he won’t resign, the PM should sack him.”
Lib Dem leader Ed Davey said: “Matt Hancock is a terrible health secretary and should have been sacked a long time ago for his failures.
“This latest episode of hypocrisy will break the trust with the British public. He was telling families not to hug loved ones, while doing whatever he liked in the workplace.
“It’s clear that he does not share the public’s values. Rules for them and rules for us is no way to run a country.”