Kane was subbed off at half-time after twice going down in the first 45 minutes.
"Two ankles, the first was a bad tackle, it was Thiago [Alcantara], the second one I didn't know well. But two injuries in both ankles, the second one worst than the first one," Mourinho said after the match.
"A few weeks," Mourinho said about how long Kane would be out. "I don't know. There are some players you can't replace."
With a busy period coming up, starting with Sunday's trip to Brighton and Hove Albion and then a home clash with Chelsea next week, Kane's absence would be keenly felt.
Liverpool centre-back Joel Matip also suffered an ankle injury in the match, deepening an already worrisome injury crisis at the back for Jurgen Klopp's men who entered the match without regulars Virgil van Dijk and Joe Gomez.
"If you have a center-half available who is a reasonable price, and who is of the quality we need, send me a message and I will go for it. It's not about [just signing a player]," Klopp said of possibly signing reinforcements ahead of the winter transfer deadline.
"It's about doing the right stuff, we can't just bring a body in, that doesn't help. Bodies, we have -- we have good kids, Nate [Phillips] did exceptionally well, Rhys [Williams] has done well when he's played."