The French goalkeeper suffered the problem in Sunday's 1-0 win over Manchester City and will spend a sustained period on the sidelines.
Scans showed he sustained ligament damage following a challenge late in the game, though Lloris is not expected to need surgery.
Fraser Forster will now step up to the number one position as Spurs enter a pivotal part of their season over the next few weeks.
They visit Leicester in the Premier League on Saturday before resuming their Champions League campaign next week against AC Milan, with London derbies versus West Ham and Chelsea also coming up.
Forster, who signed from Southampton on a free transfer last summer, has played just four times so far and made only one Premier League appearance - in the Boxing Day 2-2 draw with Brentford.
Lloris' absence will be a blow, even though he has been culpable for a number of mistakes this season.
He has made four errors that have led directly to goals - the most in the Premier League - and will be disappointed with a number of other saves he failed to make in recent games.
The 36-year-old, who has recently retired from international football with France following the World Cup, has been a mainstay of the Spurs side since last suffering a bad injury in October 2019.