Writing on Twitter, the goalkeeper said: "Thank you very much to Real Madrid for such a noble gesture. Acts like this reflect very well on football. A hug."
Bravo attached a picture of the letter, which read: "On behalf of all those who form Real Madrid, especially our president, Florentino Perez, I want to send you our best wishes for a speedy recovery.
"We are very sorry about what happened, as injuries are the worst part of football, but we have no doubt that, with your professionalism and commitment, you will continue to give many joys to football fans.
"With all our respect and admiration."
Bravo has enjoyed a storied career, winning two La Liga titles at Barca, along with two Copa Del Reys, the Champions League, the UEFA Super Cup and the Club World Cup. At City, Bravo has won the Premier League and the EFL Cup.
The 35-year-old served as City's cup goalkeeper last season, playing every game as they won the EFL Cup, and also playing in their three FA Cup encounters. He played three times in the league and once in the Champions League.