The 33-year-old defender's current deal expires in the summer, though there is still doubt about what will happen with the Premier League season set to extend beyond 30 June.
Tottenham boss José Mourinho was keen for Vertonghen to stay but suggested the defender's agent was playing hardball over a fresh contract.
Vertonghen, who joined Spurs in 2012, has not ruled out staying, but suggested back in April that he would like to play in Italy or Spain and has said there have been "serious" offers for him.
The Times claims that the Belgium international has attracted interest from Fiorentina after Spurs told him 'that he will not be offered a deal for next season'.
The club has, however, offered him the same terms until the end of July if the defender is willing to extend his stay in north London for another month.