The 31-year-old has two years left to run on his deal with the Toffees, but has broken off talks amidst rumours he could be angling for an exit from Goodison Park.
Osman has been a consistent Premier League performer for some time now and his efforts have not gone unnoticed, with England boss Roy Hodgson handing him his first international cap against Sweden last year.
According to the Metro, Osamn is contemplating a change of scenery and could turn down the offer of a one-year contract extension at Everton.
The paper claim Spurs could offer £6 million which may tempt Everton into selling up given his contract situation.
Spurs have been linked with a host of top-level midfielder operators, and have been credited with an interest in the likes of Joao Moutinho and Christian Eriksen.
While Osman wouldn't be a direct replacement should Spurs be unable to land either of those players because of his age and profile - a move for Osman would represent an attempt by manager Andre Villas-Boas to add more quality to his midfield.
It could also spell the end for the likes of Tom Huddlestone and Jake Livermore.