The 21 year-old, after a superb first season in Spain, has fielded an approach from United in the past month as they seek a replacement for wantaway England keeper Dean Henderson.
However, Marca says Spurs have also made contact.
Tottenham see Mamardashvili as a future successor for Hugo Lloris, 35.
The Georgia international's contract with Valencia extends to the summer of 2024. The Spanish club are said to be in need to sell players this summer to balance the books.