Rose was replaced by Ben Davies late in the first half at the Stadium of Light following a 50-50 challenge with Billy Jones.
The 26-year-old will undergo a scan to assess the extent of the injury on 2 February, but Spurs officials fear he has suffered medial ligament damage and expect him to be out of action for the whole of February, according to the Sun newspaper.
If confirmed, the England international will be unavailable for Premier League games against Middlesbrough, Liverpool and Stoke City, as well as the FA Cup fifth-round clash against Fulham and both legs of the Europa League tie against Genk.
"We need to assess him tomorrow," Tottenham manager Mauricio Pochettino told the club's official website after the Sunderland match.
"We hope it's not a big issue but he felt in a tackle that he twisted his knee. We'll see tomorrow."
Midfielder Mousa Dembele also picked up a knock in the Sunderland game following a heavy challenge from Jack Rodwell, while striker Heung-Min Son sustained a shin injury.
Spurs forward Harry Kane criticised Sunderland's heavy-handed tactics and called for greater protection from referees.
"You could say the other team were physical. I think there were some decisions on that pitch that were very surprising," the England international was quoted as saying by Sky Sports.
"I think Mousa getting kicked knee-high on the pitch... if you go up to anyone on the pitch and kick them knee-height on purpose, it doesn't matter if it's a 'tactical foul', let's say, it's a red card.
"There's another one. Sonny's got a big gash going down his shin where the guy's kicked the ball and followed through onto his shin. Yes, it's physical, but then I think we have to be protected as well, so obviously we're disappointed from that side of it. It was a disappointing night."