Everton made an approach for the 28-year-old last week, but it was turned down by the Cottagers who were keen to hang on to Wilson until the end of the campaign.
Wilson is out of contract at the end of the season and attracted interest from serval Premier League clubs over the past month as they attempted to lure him away from West London.
Leeds were keen on Wilson over the summer, and it looks like the offer from Everton has been enough to trigger Fulham to begin contract talks as reported by Standard Sport.
Wilson has made 173 appearances for Fulham, scoring 34 goals and registering 43 assists for the club and losing him for free would be devastating given his quality.
The report states that manager Marco Silva is hopeful that the club can come to an agreement with Wilson, a player he signed from Liverpool back in 2021 in what was his first signing.
Having been at the club for 5 years, Wilson may be ready for a new challenge, but the club will be reluctant to allow him to leave for free once the season ends.
The club have already had Rodrigo Muniz, Kenny Tete, Tom Cairney and Josh King put pen to paper this season and as talks begin with Wilson he may be the next name on the list even if it is a short term deal.