The second leg of their Champions League tie against City is on Wednesday, 17 April at Etihad Stadium.
"He was so upset after the game, it was a hard moment for him", Tottenham boss Pochettino said ahead of his side's home match against already relegated Huddersfield on Saturday.
For Tottenham to recreate their White Hart Lane fortress, they need Son Heung-min and Christian Eriksen to shine in the absence of Kane and Alli.
"Football moves so quick and after the game when Harry got injured everyone believed Sonny was going to step up and is going to be the man, but we have seen today Lucas got a hat-trick and it was an unbelievable afternoon for him", Pochettino said.
However on Friday Pochettino insisted: 'It's a stat that means nothing, Harry is one of the best players in the world and of course we're going to miss him. He needs to see the specialist next week and then we will see how his rehab goes.
"It is a little bit sad but he is looking forward to recovering as soon as possible".
The important midfield pair both played a role in Spurs winning the quarter-final first leg clash 1-0 last week and the travelling support will have their fingers crossed the duo will be available for the return game at the Etihad Stadium.
"The goal didn't come, but I though he had a big part to play in the game". There is a lot of coincidence but we are talking about an accident.
"I practise all the time and I'm so happy today because I did very well, good passes, good finishes and if we play like this as a team we can win a lot of things".
"At some point I think they would think about doing something surgically to stabilise the ligament, which is quite feasible and should restore the problem happening in the long term", Mr Davies told Press Association Sport.
Dele Alli fractured his left hand during Tuesday's Champions League victory over City.
"Players need to have the possibility to play to show the quality and that is why sometimes all the managers are disappointed when the people say about rotation, about selection of the starting XI".
There was no further addition to the scoreline in the first period, but there so almost was two minutes after the break when Juan Foyth clipped a ball in to Llorente in the area, who superbly controlled and shot on the turn but his effort smacked against the crossbar. Yesterday he saw the specialist. I am not optimistic but we'll see.