image copyrightGetty Imagesimage captionMatt Hancock was pictured leaving Downing Street with Gina Coladangelo on 1 May
Matt Hancock will not comment on reports of an extra-marital affair with a close aide, friends of the health secretary say.
The Sun published pictures of the minister embracing Gina Coladangelo, whom he made a non-executive director of the health department.
A government spokesman said the appointment had been made "in the usual way" and no rules had been broken.
Transport Secretary Grant Shapps called it a "personal" matter.
A Labour spokesperson said: "Ministers, like everyone, are entitled to a private life. However, when taxpayers' money is involved or jobs are being offered to close friends who are in a personal relationship with a minister, then that needs to be looked into.
"The government needs to be open and transparent about whether there are any conflicts of interests or rules that have been broken."
Ms Coladangelo – who is also marketing and communications director at fashion and homeware retailer Oliver Bonas – was appointed to the Department of Health's board of non-executive directors last September.
The role comes with a £15,000 salary and involves 15-to-20 days of work per year.
Other board members include England's Chief Medical Officer Chris Whitty, health minister Edward Argar and Natural History Museum director Doug Gurr.
The Sun said its picture of Ms Coladangelo kissing and hugging Mr Hancock, a 43-year-old father of three, had been taken inside the department last month.
A friend of the health secretary told the BBC he had "no comment on personal matters", adding: "No rules have been broken".
And a Department of Health spokesman said: "This appointment was made in the usual way and followed correct procedure."
Mr Shapps told BBC Radio 4's Today programme: "I think there has been for a long time a complete difference between what people do in their job… and what they do in their personal lives."
On Ms Coladangelo's appointment, he said: "Some process will have had to have been followed. I don't think there's any other way of appointing people."
Mr Shapps also told Sky News that 43-year-old Ms Coladangelo, who met the health secretary at Oxford University, would have gone through an "incredibly rigorous" process to get the job.
Ms Coladangelo, 43, is married to Oliver Bonas founder Oliver Tress, while Mr Hancock's wife Martha is an osteopath.
Asked on LBC radio whether the health secretary had been "ignoring social distancing" with Ms Coladangelo, Mr Shapps replied: "I'm quite sure that whatever the rules were at the time were followed.
"You'll recall that there was a point at which social distancing rules were changed but, as I say, I don't want to comment on somebody else's private life – that is for them."
The government's guidelines for England on hugging friends who are not part of your household states that people should exercise "personal judgement".