Sky Sports says the Hammers have been repeatedly linked with the talented centre-back following the Swans' relegation to the Championship.
And bigwigs in east London are still keeping tabs on the 24-year-old.
It's suggested discussions between the two clubs have been held but Burnley are now also in the hunt.
Mawson has been a standout performer for Swansea since signing from Barnsley two years ago.
The London-born lad has racked up 65 Premier League appearances for the Swans, scoring six goals.