West Ham had a bid for Cairney of £18m rejected in January but they may be able to lure the 27-year-old away in the summer if Fulham's promotion bid falters. The Mirror claims that Watford and Newcastle will rival the Hammers for Cairney's signature, with a £20m fee rumoured.
Young striker Ryan Sessegnon, 17, and defender Ryan Fredericks, 25, are also likely to leave Craven Cottage this summer unless Fulham get promoted.
The west Londoners have won 10 of 13 league games since the turn of the year to emerge as the chief challengers to runaway leaders Wolves and Cardiff. The Bluebirds, 2nd in the table, are currently seven points clear of the Cottagers with a game in hand.
Cairney's season has been disrupted by a recurring knee complaint, but he has still managed to chip in with two goals and some dynamic performances as Fulham seek to return to the Premier League for the first time since 2014.
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Cairney joined Fulham from Blackburn in 2015 for £3m, a fee which he has more than paid back with his performances since. He was named the club's player of the season last year as Fulham lost in the play-off semi finals, and was also voted EFL Player of the Season at the London Player Awards.
If Fulham do return to the Premier League, it will not be Cairney's first experience at that level. He played 11 games for Hull in his first senior season when the Tigers were relegated from the top flight in 2009-10.