The Hammers are said to have made a double swoop for the goalkeeper and centre-back, as they attempt to take advantage of the Swans' relegation to the Championship.
According to Wales Online, however, Swansea are standing firm and will not offload two of their prized assets on the cheap.
West Ham fell 'significantly short' of the amount City are holding out for, it is claimed, and the Welsh outfit are now bracing themselves for fresh bids.
Mawson is understood to have been the subject of a £25m offer by West Ham in January, with manager-at-the-time Carlos Carvalhal suggesting that he is worth double that amount.
Fabianski, meanwhile, is also valued highly and could see his stock rise further during this summer's World Cup with Poland.