The Daily Express says Arsenal were first linked with a move for the Kingston-born defender last year, while Chelsea are perhaps the frontrunners for Shaw's signature given he revealed on Twitter recently that he supports the Blues.
Andre Villas-Boas has hinted that Spurs could match Chelsea's £15 million valuation for him this summer.
"Any player who makes a breakthrough at the age that he has is absolutely unbelievable," Villas-Boas said.
"He's a wonderful footballer, for sure, but it doesn't mean there is direct interest from Tottenham in the player. Transfer targets will be discussed in the summer, and for now it is all speculation."