The People has printed a picture allegedly of the Sky newsreader embracing his make-up artist Camilla Tew.
Murnaghan, also known for hosting BBC quiz show Eggheads, has four children with Maria Keegan, to whom he has been married for 23 years.
The pictures, taken last week, appear to show Murnaghan on a secret date with Tew in London's Hyde Park.
A supposed onlooker told the publication: "They looked like any other loved-up couple enjoying each other's company.
"They were kissing and Dermot was being very affectionate. He was touching her, stroking her, holding her hand."