The 26-year-old France international was carried from the field at St James' Park on a stretcher with his left leg in a brace during Sunday's 2-0 Barclays Premier League victory over the Magpies with manager Manuel Pellegrini later revealing he had damaged his medial ligament.
Nasri, who was due to undergo a scan on Monday, tweeted: "Thank you everyone for your kind words and overwhelming support, I suppose its kind of good news that I will be out around 8 weeks."
Nasri was hurt in a 75th-minute challenge by compatriot Mapou Yanga-Mbiwa, which earned the Newcastle man a yellow card.
However, Pellegrini later claimed Yanga-Mbiwa should have been sent off for the tackle.
He said: "It's very serious. It was a very unfair kick and the player of Newcastle, it was directly a red card. I don't understand why he was not sent off."
Magpies boss Alan Pardew defended his player and wished Nasri a speedy recovery.
He said: "He made a point in his mind that he didn't want him to get away, and he was a bit rash in that challenge. He is not like that, though, and I hope Nasri is okay."
Nasri's injury marred a good day for City on Tyneside as they left St James' with a 2-0 victory which took them back to the top of the table.