The Jacks have reportedly priced Jefferson Montero out of a move to their Premier League rivals.
Montero endured a disappointing 2015-16 campaign, in which he failed to score a single goal and made just two assists in 23 top flight outings, and appeared as a substitute in the Jacks' 1-0 win over Burnley on Saturday.
Despite this, the Sun claims both Sunderland and Everton have shown an interest in the Ecuadorian, who Swansea are allegedly desperate to keep.
Francesco Guidolin's side are said to have already priced Montero out of a move by valuing him at £12 million, and will now offer him an improved deal to reflect that, despite his existing contract having two years to run.
Both of Montero's listed suitors have strengthened in wide areas this month, with Adnan Januzaj joining Sunderland on loan from Manchester United and Yannick Bolasie swapping Crystal Palace for Everton in a £25 million deal.
The former Villarreal speedster, 26, has played 59 times for the Welsh side in total, scoring two goals and making another 10.