The Gunners are interested in signing Bostock, who became the Eagles youngest ever player to turn out for the first team when he made his debut aged 15 years and 287 days.
Bostock has also attracted interest from Tottenham and Chelsea, while Spanish giants Barcelona are also keen and are awaiting developments.
Palace chairman Simon Jordan is said to be furious with Bostock, who could leave the club for minimum compensation if he does not sign a new deal with the Eagles.
Bostock is highly-rated by the Gunners, who would be happy to take a tribunal valuation to settle a fee if Bostock cannot agree a deal with play-off contenders Palace.
Arsenal last week