The midfielder is the Swans' Player of the Year, and they have become frustrated with the reports claiming that the Toffees are close to launching a £35m bid for the player.
"Swansea will not be pushed around and are not looking to sell," a source close to the Swansea board told Sky Sports News HQ.
"They have a figure in mind that they feel represents Sigurdsson's worth in the market and if nobody reaches that they are very comfortable about keeping him."
The Iceland international was ever-present for the Welsh outfit last season, playing all 38 Premier League games and scoring nine goals along with 13 assists.