The 22-year-old has returned to Stamford Bridge after experiencing a difficult loan spell with Bundesliga side Bayer Leverkusen last term.
Hudson-Odoi started just 13 of his 21 competitive appearances for the German club, registering only one goal and one assist.
After failing to make an impression in Germany, Hudson-Odoi is looking to find a club where he can enjoy a regular run of games.
With the player's contract set to expire in 12 months, Chelsea are prepared to part ways with their academy graduate this summer.
Nottingham Forest and Lazio have both been linked with a move, but it appears that Fulham are leading the race to secure his signature.
A recent report suggested that the attacker has reached an agreement over personal terms with Marco Silva's side.
With Hudson-Odoi seemingly keen to complete the switch to Craven Cottage, the two clubs are currently discussing a transfer fee.
According to journalist Fabrizio Romano, Fulham have made a verbal proposal of £4m, which falls below Chelsea's £8m valuation.
Despite the difference in asking price, the Cottagers are confident of getting a deal over the line with negotiations believed to be at an advanced stage.
Fulham are determined to bolster their wide options following the departures of Neeskens Kebano and Manor Solomon this summer.
Hudson-Odoi, who has earned three caps for England, has made 126 appearances as a Chelsea player, scoring 16 goals and providing 22 assists.
The Cottagers have recently stepped up their transfer business, signing Mexico international Raul Jimenez from Wolverhampton Wanderers.
They are also on the verge of unveiling Calvin Bassey as their latest addition after agreeing a €21m (£18m) fee with Ajax.