The Hammers had a double bid rejected by Swansea that also included goalkeeper Lukasz Fabianski but remain keen on Mawson having missed out on Jonny Evans to Leicester City.
Should Spurs join the race?
Tottenham Hotspur are also in the market for a new centre-back, as reported by the Evening Standard, as they seek a replacement for Toby Alderweireld.
Alderweireld is strongly linked with an exit from the club with Manchester United the Belgian's most likely destination and Newcastle United captain Jamaal Lascelles and Ajax's Matthijs de Ligt are being considered.
However, it will take an offer in the region of £45m to convince the Magpies to sell Lascelles, while the Independent reports that De Ligt could cost as much as £44m.
With Mawson available for less than half of both of those valuations should Tottenham consider returning for a defender they were linked with by the Guardian only last month?