The 24-year-old had an operation late on Monday night after sustaining the problem in City's 1-0 Barclays Premier League win over Swansea.
It had been thought that Richards could be back in action within six to eight weeks, but the academy graduate told the club website it could take much longer.
"The positive thing is that the injury has been repaired so the knee will be as good as new," he said.
"I have a brace on it and I am using crutches. I can't do anything at all for the next four to six weeks and I am probably looking at 12-16 weeks before I am back in action."
The injury has also left Roberto Mancini sweating over his selection problems at right-back.
Both Pablo Zabaleta and Maicon missed the Swansea game with hamstring and foot problems respectively.
City are hopeful that both players are close to a return, with Zabaleta optimistic of being available for the weekend trip to West Ham.
But should there be any setback, Mancini would be left with a major headache.
Kolo Toure replaced Richards on Saturday, but the Ivory Coast international is not registered in City's Champions League squad.
City must beat Ajax next Tuesday to retain any hope of going through from the group stages, and defeat would likely also end their chances of even taking third position in Group D and a place in the Europa League.
The injury comes as a personal blow for Richards in what has been a frustrating season so far. He missed the start of the campaign with an ankle injury suffered while playing for Team GB at the Olympics, and has played in just four of City's 13 games.
He was injured in the 85th minute of Saturday's game, twisting awkwardly and crumpling in obvious pain before being taken off on a stretcher.
"I knew straight away in the game that it was quite a bad injury," Richards said. "The knee locked and I couldn't straighten it.
"I suppose it could have been even worse and not repairable which would have meant removing the meniscus; so I am being positive. Injuries are all part and parcel of the job but this coming on top of the other one I do feel a little hard done to!
"I will be working as hard as I can to get back as quickly as I can and I thank all the fans who have sent their best wishes."
However, Richards has backed his team-mates to succeed without him, having overcome some indifferent early season form to rise to second in the Barclays Premier League table.
Asked about his own displays since returning, he said: "I felt I was doing okay. We beat Sunderland, West Brom and Swansea and kept two clean sheets in those three games, so things were going well in the Premier League.
"I am sure the lads will continue that run of form and I will be looking forward to getting stuck into the title defence as soon as possible when I return. Everyone has their opinion on how we have started the season but the league table looks pretty good!"