It was confirmed last week that Kane needed surgery on his left hamstring and will be out of action until at least April after sustaining the injury on New Year's Day when scoring a goal against Southampton that was ruled out for offside.
The 26-year-old will be hoping to be fit for Spurs' end-of-season run-in and Euro 2020 duties for England.
Ahead of Tuesday's FA Cup third-round replay against Middlesbrough, Mourinho was asked for an update on Kane's fitness and whether he would need another striker to come in, having previously been linked to Milan's Krzysztof Piatek.
Mourinho said: "We try the best we can. News on Harry we don't have and if you ask me every time we come here the answer will be the same.
"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.
"I think this market in January is a strange market. Sometimes the opportunities are there and sometimes they are not there.
"Sometimes clubs are ready to sell or loan and sometimes club are not. It's not an easy one. It is just the window of opportunity and sometimes the opportunity is there and sometimes it is not."
Mourinho feels Spurs' injury concerns, with Ben Davies (ankle) the first to be struck down during the Portuguese's tenure, are making it difficult to make long-term plans.
"If you want to know what I wanted to do here, you look to my first game, against West Ham. That's the way I thought I was going to develop the team. Defend with a back four, attack with a back three.
"That means that my left-back defends as a left-back and then when we go in attacking organisation, my left-back becomes one of the three in the back. Build with three, protect the right-back and do overloads on the other side.
"That was the way I wanted to develop the team. That was the first day of unlucky, goodbye Ben Davies and then I think everything started with Hugo Lloris.
"You lose your captain for so long and then in my case I lose Ben. We tried but then the players are different and then you lose one more, and you lose one more, and you lose Kane, and you don't have a striker.
"You go to Middlesbrough and you play with Lucas [Moura] and Sonny [Son Heung-min] both up.
"But then you realise that the team is comfortable with width and then you have to bring one to the side ... instead of building something that you see, the puzzle is coming and we are developing."