The 26-year-old is expected to be part of a mass Elland Road exodus following their relegation from the Premier League and was initially thought to be close to joining Unai Emery's side.
However, according to The Athletic, Everton are now poised to sign the former Manchester City man, who was undergoing a medical with the Toffees on Sunday.
Sean Dyche's side have supposedly triggered a clause in Harrison's deal allowing him to join on loan for the 2023-24 season, and his move would likely be made permanent next year.
Harrison had already agreed personal terms with Everton and is set to leave Leeds after a productive five-year spell, although he only joined permanently in 2021 following a three-year loan.
The former England Under-21 international has 34 goals and 32 assists to show from 206 matches for the Whites, but he has missed the start of their 2023-24 season through injury.
Meanwhile, West Ham United are reportedly interested in a deal for Udinese midfielder Lazar Samardzic, who has also been watched by Inter Milan.
The Hammers have already captured Edson Alvarez to replace Declan Rice but are now at risk of losing Lucas Paqueta, who is a prime target for Manchester City.
Samardzic posted a phenomenal set of numbers at youth level for Hertha Berlin, scoring 40 goals in 40 games for their Under-17s, but he made just nine appearances during an unsuccessful spell with RB Leipzig.
Since joining Udinese in 2021, the 21-year-old has scored seven goals and set up seven more in 63 matches for Le Zebrette, including five goals and four assists from 37 games in the 2022-23 Serie A season.
According to the Daily Mail, West Ham are ready to enter the race to sign Samardzic, and the Hammers could land the Serbia international for a fee of around £22m.
Inter were thought to be the frontrunners for Samardzic's signature, but it has now been claimed that the two clubs have reached an impasse over personal terms, potentially jeopardising his San Siro switch.
Finally, Middlesbrough striker Chuba Akpom is reportedly on the verge of sealing a €14m (£12.1m) transfer to Dutch giants Ajax.
The former Arsenal product was in prolific form for Michael Carrick's team last season, notching 28 goals from 38 Championship games as Boro lost to Coventry City in the playoff semi-finals.
Akpom stormed to the Golden Boot with his 28-goal haul, while also being named the Championship Player of the Season, and his displays have naturally attracted glances from top-flight teams.
Ligue 1 side Lens were believed to be considering a move for Akpom as a potential successor to Lois Openda, who left the Stade Bollaert-Delelis to join RB Leipzig earlier this summer.
However, The Athletic now claims that the 27-year-old is set to sign for Ajax, who will pay an initial €12m (£10.4m) to Middlesbrough, with an additional €2m (£1.7m) in performance-related add-ons.
Akpom - whose Boro deal expires in 2024 - is expected to complete his medical with the Amsterdam side on Monday before putting pen to paper on a five-year deal.
The Englishman had previously announced his desire to stay at the Riverside Stadium, although he admitted that his future was not entirely in his own control.