The 22-year-old enjoyed a successive loan spell at Selhurst Park in 2021-22, scoring eight goals and providing three assists in 39 appearances across all competitions.
However, the Daily Star reports that Palace, West Ham and Everton are all interested in recruiting the England international this summer.
Palace head coach Patrick Vieira has expressed his desire to work with Gallagher for a second season, but as things stand the midfielder will be departing the Eagles.
Speaking to the Daily Star, Vieira said: "Yes I think the fans love him. I love him. The players in the dressing room respect and love him.
"I think it was a really good season for Conor. What made us proud is that he came to this football club for a year and he is leaving this football club as a really different player. He's an international player now. We played our part. And he played his part as well.
"Now we wish him good luck and we wait and see what will happen. But we may have a player in the squad already who can replace him."
Gallagher still has three years remaining on his contract at Stamford Bridge. body check tags ::