Manuel Pellegrini's side are on the lookout for a new goalkeeper after Joe Hart returned to parent club Manchester City.
Their search for a new man between the sticks led them to Fabianksi, with the Pole believed to be keen on the move, having told Swansea of his desire to leave the club following their relegation form the top flight.
The 33-year-old is currently away with his country at the World Cup, but negotiations are thought to have been progressing in his absence, with deals for him and Toulouse defender Issa Diop thought to be on the verge of completion.
Fabianski has just a year left on his contract at the Liberty Stadium, but it appears his four year stay in south Wales is coming to an end.
Speaking in Russia, Fabainski admitted that any departure would come with a heavy heart, saying: "If I leave, I will regret that it happened at the moment when we fell from the league.
"I spent beautiful moments there. I remember that the club helped me in a bad time when I had health problems and I wasn't really playing at Arsenal.
"I was trusted and I met many great people in the city, I have a beautiful home there, in which I raised my son. The memories will last a lifetime."