The Belgium international, 24, changed the game from the bench but fell awkwardly in injury time.
His agent Patrick de Koster initially said De Bruyne would miss six weeks.
But, after seeing a specialist, the £55m former Wolfsburg player said: "I'll be out for around 10 weeks."
De Bruyne could miss up to 13 league and cup games, including the League Cup final with Liverpool on 28 February, both legs of the Champions League last-16 tie with Dynamo Kiev and the Manchester derby on 20 March.
The Belgian is City's second top goalscorer with 12 this season, four behind striker Sergio Aguero.
De Koster added: "Kevin told me the only thing he can do is work hard and come back. Kevin is sad. His dream is to always be playing football."
De Bruyne scored one goal and set up another to help City to a 4-3 aggregate victory over the Toffees.
Everton goalkeeper Joel Robles, who repeatedly tried to lift up De Bruyne as he lay injured, used social media to say sorry.
"I would like to apologise to Kevin de Bruyne for my reaction to his injury," said the 25-year-old Spaniard.
"In the heat of the moment I didn't realise he was badly hurt. I wish him all the best and a speedy recovery."
Manchester City midfielder Kevin de Bruyne says he will be out for about 10 weeks after injuring his right knee during Wednesday's League Cup semi-final victory over Everton.