Kane needed surgery on his left hamstring after suffering a ruptured tendon against Southampton on New Year's Day.
Tottenham have said they expect Kane to be back in training in April, with no exact date set for the striker's return to first-team action.
Mourinho was questioned about Kane at a press conference on Monday and his answer suggested there could be some doubt over the 26-year-old's participation at the European Championship in June and July.
Mourinho even appeared to admit Kane is not guaranteed to play again before next season, which would be a nightmare scenario for the Tottenham boss and England manager Gareth Southgate.
"News on Harry we don't have and if you ask me every time we come here the answer will be the same," Mourinho told reporters.
"We expect him to be out until, I don't know, mid-April, end of April, May, next season I don't know. I have no updates to give on Harry."
Kane tweeted from his hospital bed that his operation at the weekend had been a success and that he was keen to get on with his recovery.
He has a history of recovering quickly from the numerous ankle injuries that have cost him playing time in recent seasons.
Kane was the leading scorer in Euro 2020 qualifying with 12 goals and will hope to be ready in time for England's first game of the tournament against Croatia on June 14.