The Mirror says Crouch was warned he was not guaranteed a place on the 25-man Premier League squad Spurs, and the other 19 clubs, must submit by 5pm on Friday.
The 6ft 7in Crouch joined the north London side from Portsmouth for £9million three years ago and played on Sunday in the 5-1 home defeat by Manchester City, but the arrival of Emmanuel Adebayor on loan from City has changed his prospects.
Stoke made their first move for Crouch and Wilson Palacios earlier in the summer , but the big forward was determined to remain with the capital club.
Palacios eventually agreed to move north earlier on Wednesday, but even after Potteries boss Tony Pulis returned with an improved four-year contract for Crouch, he once again rejected the move.
However, with the warning from Levy ringing in his ears, Crouch changed his mind on Wednesday night and agreed the move to the Britannia, where he will be able to play Europa League football.