The Irons are in the market for a new stopper after Joe Hart and Adrian failed to nail down a starting spot in the first team last season.
Hart's disappointing showing will likely mean he doesn't join the club on a permanent basis, and Spaniard Adrian's future is uncertain after he voiced his displeasure about being rotated.
Poland international Fabianski had a relatively good season at the Liberty Stadium despite suffering relegation with the team, and is expected to move on this summer.
As reported by football.london, he is the subject of interest from West Ham, but the London club only bid £5m which was swiftly rejected by the Swans.
It would appear they believe their relegation could mean a cut-price deal for Fabianski, but other clubs are expected to come in for him and that offer is surely way off the mark.
Fabianski has proven himself to be a dependable goalkeeper in the Premier League over the past few years. To date, he has made almost 200 Premier League appearances across stints with Arsenal and Swansea.
Meanwhile the Hammers are said to be closing in on Lazio star Felipe Anderson, which would be a huge coup for the club and could potentially mean a replacement for Dimitri Payet.