In a light-hearted video response captioned “Rather takes the veneer off the story”, Mr Hancock said: “So I just got back to the office to find a journalist has called to ask if I went to Istanbul at the weekend to have hair transplants and veneers.
“This story has no teeth. It’s hair today and gone tomorrow.”
Istanbul is popular for its affordable cosmetic work, with many tourists travelling there for hair transplants and dental appointments.
Mr Hancock, who announced in December that he would stand down as an MP at the next election, recently closed his app and urged people interested in what he was doing to follow him on TikTok.