The Mirror says Bostock should sign his first pro deal at Palace on his 16th birthday in January despite the Stamford Bridge outfit being prepared to pay him a fortune and also give Palace a record £2million for a talent of his age.
Bostock has become Palace's youngest-ever player and his family thinks he should progress with his local team rather than move.
His dad is a Palace season ticket holder and thinks that his son will rise quicker at Selhurst Park - although there are tempting offers elsewhere.
Boss Neil Warnock, who has not picked him for Palace's last three games, said: "John has done ever so well but it's difficult to maintain that at 15. He has got his exams and all sorts happening in his life."