The Cottagers stood firm when West Ham lodged two bids for Cairney in January but their pursuit is intensifying as the club's owners look to make good on their promise of an improved team for next season.
Sky Sports News reports that Fulham are unwilling to sell at any price, but it remains to be seen what Cairney will make of the offer, as he was reportedly keen to join West Ham in January.
He has since scored the winning goal at Wembley in the Championship playoff final that saw Fulham return to the Premier League, and it remains to be seen whether their promotion will convince him to stay on at Craven Cottage.
The Hammers have already raided Fulham for defender Ryan Fredericks and Slavisa Jokanovic will not want to lose another of his key players before the start of the 2018/19 season.
As well as Fredericks, Pellegrini has also signed Issa Diop from Toulouse and Polish goalkeeper Lukasz Fabianski from Swansea City.
Cairney scored six goals last season, his lowest tally in three years at Fulham, but his is a contribution which cannot be measured by statistics alone. He is an important player under Jokanovic and it would be a hammer blow if Fulham were to lose him.