The Mirror claims Baggies boss Alan Pardew wants to bolster his midfield ranks with the addition of the ex-Hull City starlet and Albion are prepared to pay up to £15m for Cairney.
Their opening approach of £12million is understood to have been to ‘test the water’ for a player who Fulham value very highly.
The 26-year-old was the subject of a £20million bid from Newcastle last summer, which the west London club rejected.
But the Cottagers could cash in on Cairney now to raise the funds to buy a striker later on in the January transfer window.
Fulham managed to beat promotion rivals Middlesbrough on Saturday without Cairney who was only named on the bench, having been struggling with a recurrence of a knee injury.
The Cottagers skipper has made 17 appearances in the Championship this term, scoring just once.
Watford boss Javi Gracia does not think Abdoulaye Doucoure's form will be affected by his omission from France's latest squad.