The Toffees are willing to offload Gray this summer after completing the signing of his replacement Arnaut Danjuma, who has joined the club on a season-long loan deal from Villarreal.
According to The Sun, Fulham are interested in signing the winger and are hoping to agree a £7m transfer with Everton.
However, the report claims that Everton have set an asking price of £10m, with 12 months left to run on Gray's contract, which includes an option to extend by an additional year.
The Cottagers are looking to strengthen their wide options following the departures of Neeskens Kebano and Manor Solomon this summer.
Gray, who has also attracted interest from Al-Hilal and Crystal Palace, registered six goals and one assist in 36 appearances last term.
As for West Ham United, they have identified Ajax midfielder Edson Alvarez as a replacement for Declan Rice, who recently completed a move to Arsenal.
After being priced out of a move for Fulham Joao Palhinha, the Hammers are now set to turn their attention to other midfield targets.
According to Football Insider, David Moyes's side are prioritising a move for Alvarez, with discussions taking place with his representatives.
Alvarez looked set to complete a transfer to German club Borussia Dortmund earlier this summer before the move fell through.
The Mexico international, who can play as a defensive midfield and a centre-back, is under contract with the Dutch giants until the summer of 2025.
Since joining Ajax from Club America in 2019, Alvarez has featured in 147 games across all competitions, chipping in with 13 goals.
Over in France, Marseille have reportedly agreed a deal with Watford to sign Senegal international Ismaila Sarr.
After joining Watford from Rennes for a club-record fee of £28.8m in 2019, Sarr has gone on to score 34 goals and provide 24 assists in 131 appearances.
With 12 months left to run on Sarr's current deal, the Championship side are now set to cash in on the 25-year-old winger.
According to journalist Fabrizio Romano, Sarr is on the verge of completing a move to France after Marseille reached an agreement with Watford.
Sarr, who has earned 54 caps for Senegal, is expected to undergo a medical on Monday ahead of the proposed transfer.
Should everything go according to plan, the attacker will put pen to paper on a long-term contract until the summer of 2028.
Meanwhile, Brighton & Hove Albion have reportedly given James Furlong permission to leave the club on loan this summer.
Furlong, who joined Brighton's academy in 2019, is highly regarded by the Seagulls despite making just one senior appearance for the club.
The left-back spent the second half of last season on loan at Scottish Premiership side Motherwell, where he featured in 16 competitive games.
Since returning to Brighton, Furlong has put pen to paper on a new contract, extending his stay with the club until the summer of 2025.
According to Football Insider, the Seagulls are prepared to let Furlong leave on another loan deal, amid interest from a number of Championship and League One clubs.
Roberto De Zerbi's side believe that a temporary move will benefit the 21-year-old and allow him to gain regular playing time.