The 29-year-old tore ligaments during training before Belgium's Euro 2016 quarter-final against Wales in Lille on Friday.
Belgium manager Marc Wilmots said: "It happened all alone this morning in the last minute of training."
Vertonghen looks set to miss the early part of the Premier League season, which starts on 13 August.
He played in Belgium's three group games at Euro 2016 as well as Sunday's 4-0 win over last 16- opponents Hungary.
Wilmots added: "I was about to whistle for the end of training, he received a ball in front of me and fell to the ground clutching his left ankle."
Vertonghen said: "Gutted to say that my Euros is over. Good luck to my team-mates for the rest of the competition."
Wales manager Chris Coleman said: "He is a good player, an outstanding performer. But they have so much talent."
Vertonghen played at left-back for Belgium at Euro 2016 but was an option to move to centre-back against Wales to cover for Thomas Vermaelen's suspension.
Jason Denayer may partner Toby Alderweireld in central defence, with fellow 21-year-old Jordan Lukaku at left-back.
"I still have a few alternatives to replace Jan," said Wilmots. "I will think it through and we will decide tomorrow."
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Belgium and Tottenham defender Jan Vertonghen has been ruled out for up to eight weeks with an ankle injury.