The Sun says Spurs will make a £15 million swoop for Mawson.
Boss Mauricio Pochettino has been in touch with the Welsh club over the England Under-21 defender.
Spurs see him as part of their squad for next season and eventually a long-term starter at White Hart Lane.
Mawson has been a huge hit for Swans since moving from Barnsley in a £5m deal at the start of the campaign — and his value has now trebled.
It is a massive turnaround for the 23-year-old, who left Brentford a couple of years ago for a pittance. But he could be heading back to London as a major Premier League capture.