The 33-year-old former Arsenal custodian, who joined the Swans in summer 2014 and has made 150 appearances, still has a year remaining on his contract at the Liberty Stadium.
However, with the Swans facing Championship football after being relegated from the Premier League on Sunday, Fabianski could look to move on to remain in the top flight next season.
Newcastle have been linked, but Palace boss Roy Hodgson is keen to bolster his ranks with more competition for Wayne Hennessey as Julian Speroni is approaching his 39th birthday on Friday and Diego Cavalieri was only brought in as cover on a deal until the summer due to an injuries.
If Swansea agree to offload Fabianski the player will have to decide where he would be given more chance of regular first-team football.
Watford defender Sebastian Prodl will face world class opposition from Russia, Brazil and Germany after being called up by Austria.
Huddersfield have lost out to Premier League rivals Brighton for the free transfer signing of Mainz's Nigerian defender Leon Balogun.
Jean Michael Seri will be allowed to leave Nice in the summer transfer window, president Jean-Pierre Rivere has confirmed.
Captain Marek Hamsik has admitted that he could leave Napoli this summer as he has been linked with a big-money move to China.