Alderweireld, who suffered the injury during the first half of Tottenham's Champions League victory against Real Madrid, was already ruled out of Saturday's Premier League derby at Arsenal.
The severity of his injury, however, is an unexpected blow.
"He's out until after Christmas," Pochettino told reporters. "He feels disappointed but he's trying to focus on his recovery and the first stage of the rehabilitation.
"Always with this kind of injury we have to be cautious."
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Harry Kane, Dele Alli and Harry Winks, who all withdrew from the England squad for friendly internationals against Germany and Brazil, trained on Thursday, although Pochettino refused to say whether they would all play.
"We need to assess them again tomorrow," he said.
Tottenham are third in the table, four points above Arsenal, but are yet to win at the Emirates since Pochettino took over in 2014, all three games ending in 1-1 draws.
(Reporting by Martyn Herman; Editing by Toby Davis)
