The 24-year-old had been close to a 6 million euro move to Fiorentina, but Spurs have moved quickly to hijack the deal as they look to add to the five new arrivals already brought in by Mauricio Pochettino.
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Stambouli, who came through the youth system at the Ligue 1 club, has just 12 months remaining on his existing contract and now looks set to follow former teammate Remy Cabella to the Premier League.
"An agreement has been reached between Montpellier and Tottenham to transfer Benjamin Stambouli," read a tweet on the French club's official feed.
The potential arrival of Stambouli, a tough-tackling defensive midfielder, could signal that Spurs have admitted defeat in their pursuit of Southampton's Morgan Schneiderlin.
Pochettino was interested in securing the services of the Saints star and was prepared to make him his most expensive arrival this summer, but manager Ronald Koeman has been vocal of his reluctance to sell.
The Montpellier man has made 14 appearances for the France Under-21 side, though has yet to earn a call-up to the senior national team.