Playmaker Nasri is expected to be sidelined for around eight weeks following the challenge by Yanga-Mbiwa in Sunday's Barclays Premier League win for City at St James' Park.
The City man had to be carried off on a stretcher after being caught by a bad tackle from Yanga-Mbiwa, who was booked.
"I didn't intend to injure him. There was no malice. Unfortunately there is this consequence," Yanga-Mbiwa told L'Equipe.
There were initial fears that Nasri could be facing a longer stretch on the sidelines, a worrying scenario for the France player in a World Cup year.
"When I heard that, I was relieved," Yanga-Mbiwa said of the shorter timetable for Nasri's recovery. "I was really troubled by what happened. He's a player who is in the France team and I know what that represents."