It is understood that Toffees boss Ronald Koeman has put the Iceland international right at the top of his transfer wish-list having made failed attempts to sign him in January and the 2016 summer transfer window.
The Swans managed to convince the 27-year-old playmaker to sign a new contract last August amid speculation over his future and are therefore not in a position where they need to cash in on his services.
Everton are thought to be willing to offer £30million for Sigurdsson but they could be wasting their time if the Welsh club stay in the Premier League as they remain determined to keep him.
However, should Swansea drop out of the top flight then Everton will likely get their man as Sigurdsson is not willing to play in the Championship.
After losing at Bournemouth on Saturday, the Swans are 17th in the table, three points clear of the bottom three.