Kane went down a couple of times and required treatment in the first half but did not return for the second period, with Erik Lamela taking his place.
Tottenham failed to build on their eight-game unbeaten run as goals from Roberto Firmino, Trent Alexander-Arnold and Sadio Mane rendered Pierre-Emile Hojbjerg’s stunning effort a mere consolation.
Mourinho revealed the extent of Kane’s injury after the game, telling Match of the Day: "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 – a few weeks [out], I don't know.
"There are some players you can't replace. When it happens it happens but I believe we have to fight against it, we cannot do anything else.
Kane has scored 22 goals in 29 appearances across all competitions this season for Tottenham, who lie sixth on the league table, eight points behind leaders Manchester City.
The England captain could miss Tottenham’s upcoming games against Brighton and Hove Albion, Chelsea and Everton as a result of his latest injury.
