The 25-year-old midfielder has been with Atletico Madrid for one year after leaving Chelsea, where he was a product of the academy.
Despite starting the 2024-25 season strongly, scoring two goals in his first five games, Gallagher's performance declined as the season progressed.
Atletico Madrid values Gallagher at approximately £36 million, slightly lower than the reported transfer fee they paid for him last summer.
With only 19 La Liga starts in the 2024-25 season, Gallagher is believed to be seeking more playing time ahead of the 2026 World Cup.