The Sun published pictures of the married Cabinet minister appearing to kiss Gina Coladangelo, who the newspaper said was hired by Mr Hancock last year.
The images, which appear to be captured from CCTV footage, were taken on May 6 from the headquarters of the Department of Health and Social Care (DHSC), the newspaper added.
It said the minister hired Ms Coladangelo as an unpaid adviser on a six-month contract in March last year, before appointing her as a non-executive director at the department.
Mr Hancock, who is said to have met Ms Coladangelo at university, has been married to his wife Martha for 15 years and they have three children together.
Ms Coladangelo is the marketing and communications director at Oliver Bonas, a British retailer founded by her husband Oliver Tress.
Liberal Democrat leader Sir Ed Davey said on Twitter: “The reason Matt Hancock should resign is that he is a terrible Health Secretary, not because of his private life.
“From the PPE scandal, the crisis in our care service and the unbelievably poor test and trace system, he has utterly failed.”
Meanwhile, Labour said the Government needs to answer whether the Health Secretary had broken any rules or there had been “conflicts of interest” in the appointment of his closest adviser.
An opposition party spokesman 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.”
Mr Hancock has has also been accused of breaking his own Covid rules with the affair.
The Government’s road map out of lockdown said people should continue to keep their distance from anyone not in their household or support bubble until May 17 - 11 days after the photo of Mr Hancock and Ms Coladangelo was allegedly taken.
Defending his colleague on Friday, Transport Secretary Grant Shapps said that he would not be commenting on the “entirely personal” matter following the reports about his Cabinet colleague.
When asked if the Health Secretary had been ignoring social distancing rules when the images were taken, Mr Shapps told LBC he is “quite sure that whatever the rules were at the time were followed”.
Asked about the rules around appointing friends to Government positions, Mr Shapps told Sky News: “First of all, I think the actual issue is entirely personal for Matt Hancock.
“In terms of rules, anyone who has been appointed has to go through an incredibly rigorous process in Government, so whatever the rules are, the rules will have to be followed.
“There are no short cuts to that, as anyone who has had anything to do with the appointments system in the Civil Service knows.
“There are very strict rules in place.”
Mr Hancock was not at his north London home on Friday morning. The DHSC was contacted for comment.