The 16-year-old midfielder became Palace's youngest-ever player aged 15 years and 287 days when he featured against Watford last October.
Palace boss Neil Warnock was hoping to tie Bostock to a contract in south London but now appears to have admitted defeat in trying to keep hold of his prodigy.
He said: "I don't honestly know. I've been told he's going to sign for Tottenham but I'm not sure. I'll be speaking to his dad.
"It's disappointing really if that's the case because I've not been told myself. But these big clubs can do what they want, can't they?"