Manuel Pellegrini is keen to sure up his defence, as the East London club found themselves conceeding the joint-most goals in the Premier League last season.
Mawson, 24, made 44 appearances for the Welsh club last campaign at the centre of defence, while Polish 'keeper Fabianski was ever-present in the Swan's league side.
Pellegrini has shown great ambition so far at the helm of the 'Hammers', having already signed Fulham's Ryan Fredericks and opening talks for the signature of Brazilian winger Felipe Anderson from Lazio.
However, if the Premier League side are to nab the duo, it seems they will have to part with more money.