West Ham, Newcastle and Everton are also interested in a loan deal for Gallagher, who Chelsea have previously valued at £50million.
The 22-year-old enjoyed a brilliant season on loan at Palace last season and the Eagles hope to persuade him that a return to Selhurst Park will help his chances of being named in the England squad for the World Cup in Qatar this winter.
Gallagher has started the past two matches for Chelsea but had to apologise to his team-mates and fans after being sent off for two yellow cards in Saturday’s 2-1 win over Leicester.
He also struggled in the 3-0 defeat at Leeds but Thomas Tuchel does not want to lose him, with N’Golo Kante dogged by fitness issues and facing a month out injured.
The latest injury for Kante has left Chelsea short of midfield options and Tuchel sees Gallagher as the most like-for-like replacement for the France World Cup winner.
Tuchel also has plans to switch formation to play four at the back, which could see Gallagher played in his preferred No8 position, having mostly lined up as a No6 this season.
Amid Palace interest in Gallagher, Chelsea have expressed an interest in Wilfried Zaha.
Zaha is in the final year of his contract at Selhurst Park and has been identified as an alternative option if Chelsea do not sign Anthony Gordon.
Palace are reluctant to sell their star player so late in the transfer window but their interest in Gallagher means there could be talks between the two clubs in the coming days.