The former Bayer Leverkusen loanee is not in Mauricio Pochettino's plans for the 2023-24 season, and all parties are seeking an exit during the current transfer window.
A move to Nottingham Forest or Lazio has been mooted, but Fulham - who could offer Hudson-Odoi the chance to stay in London - are now believed to be leading the race.
According to the Evening Standard, Hudson-Odoi has now given the green light to a contract with the Cottagers and is keen to wrap up his switch to Craven Cottage permanently.
The 22-year-old is only under contract with Chelsea for another year, but the Blues are allegedly trying to spark a bidding war between Lazio and Fulham, and as a result a fee is yet to be agreed.
Hudson-Odoi has registered 16 goals and 22 assists in 126 matches for the Chelsea first team, but injuries have prevented him from living up to his potential in West London.
Meanwhile, AC Milan-bound attacker Samuel Chukwueze has been pictured arriving for his medical with the Rossoneri ahead of his switch from Villarreal.
The 24-year-old is expected to join Stefano Pioli's side in a deal worth up to €28m (£24m), and he will commit the next five years of his career to the Italian giants.
A deal should go through in the coming days, and a video shot by Calciomercato appears to show Chukwueze turning up for his medical examinations on Wednesday.
The Nigeria international was in good spirits as he was driven into the La Madonnina clinic, where he acknowledged a group of supporters before heading inside.
Chukwueze has recorded 37 goals and 31 assists in 207 appearances for Villarreal since joining the Yellow Submarine's youth academy in 2017, but his contract only runs for another season.
Arsenal and Real Madrid were previously linked with a swoop for Chukwueze, who has won the Europa League and reached the semi-finals of the Champions League with Villarreal.
Back in England, Manchester United goalkeeper Nathan Bishop is understood to be on the verge of a permanent exit from the Red Devils as he prepares to join Sunderland.
The 23-year-old joined the Red Devils' academy from Southend United in 2020 and has turned out 12 times for the Under-21s, but he has not made a single competitive appearance for their senior team.
Bishop has been offered valuable first-team minutes during pre-season and featured in a recent 3-1 loss to Wrexham, although he was booed after a collision with Paul Mullin left the striker requiring hospital treatment due to a punctured lung.
Bishop is only under contract at Old Trafford for another year, and according to the Northern Echo, a deal is now in place for the shot-stopper to move to Sunderland.
Bishop's switch to the Stadium of Light should supposedly go through once Man United return home from their USA tour, and the former England Under-20 international will arrive as a backup to Anthony Patterson.
While he has not played for Man United in a competitive match, Bishop enjoyed a successful loan spell with Mansfield Town in the 2021-22 season, helping the Stags reach the League Two playoff final before they lost 3-0 to Port Vale.
Finally, Celtic centre-back Carl Starfelt is apparently set to be the subject of a formal bid from Russian outfit Spartak Moscow.
The 28-year-old previously spent two years in Russia with Rubin Kazan between 2019 and 2021 before earning a £4.3m switch to Celtic, where he quickly became an important part of the team.
With Starfelt in the ranks, Celtic have won five trophies from six on offer during the past two seasons, and the Sweden international managed three goals and one assist from 37 games last term.
However, Football Scotland now claims that Spartak Moscow will soon test Celtic's resolve with an official bid for Starfelt, who is not due to become a free agent until 2025.
It is unclear whether Celtic have any intention of letting the 28-year-old leave, but Brendan Rodgers's side recently bolstered their defensive ranks with the signing of Maik Nawrocki from Legia Warsaw.
Starfelt boasts a total of three goals and two assists from 86 games in a Celtic jersey, and he started 19 Scottish Premiership contests on the bounce during the 2022-23 campaign.