The 23-year-old started in 27 league matches for the Welsh club last season and scored four goals in the process. His form for Swansea saw him attract interest from the north London club, according to the Telegraph.
Swansea were demanding a fee of around £30m ($38.3m) for the England Under-21 international. Mauricio Pochettino's side were ready to send Josh Onomah on loan as a part of the deal as the north London club were hoping to reduce the asking price.
Mawson revealed Tottenham never made a firm approach in securing his services and stressed that he is happy with his current employers.
"Again, I had this conversation with my friends, my family, my manager," Mawson said on Sky Sports' Premier League Daily show, as relayed by HITC.
"I don't listen to it unless there is something properly in paper. If the manager called me and said, 'Listen, this is black and white, this is what's happened, this is what's not happened', I respect that.
"Until there is a contract or until there is proof of it, then I don't ever think that. I don't think too much into it. I am a Swansea player and I am more than happy here."
Instead of signing the Swansea centre-back, Tottenham landed Davinson Sanchez as the defender became Spurs' first summer signing. The Colombian international moved to London from Ajax and signed a six-year deal with the Premier League outfit.
Meanwhile, Mawson has failed Swansea manager Paul Clement and the club's board and owners after the Swans secured the services of Renato Sanches. Bayern Munich allowed the Portugal international join the Premier League club on a season-long loan deal.
"Some of us may have thought 'is it really going to go through? Is it nonsense?' You never know so until it all got confirmed it was just 'that will be a good signing if we can pull it off'," Sky Sports quoted Mawson as saying.
"Fair play to the manager, the chairman and the owners for pulling that off. They got that one over the line and I think he's going to be a really big asset for us."