The Premier League champions were keen on bringing the Portugal left-back to Old Trafford on transfer deadline day, but their loan bid was not completed in time.
Real decided not to sanction Coentrao's departure when a possible replacement, Guilherme Siqueira of Granada, was loaned to Benfica instead.
Coentrao, 25, left Benfica for Real in 2011 and some reports in the defender's homeland claim he was upset when the move to United failed to go through.
However Coentrao, who played for his national side in Friday night's 4-2 World Cup qualifying win over Northern Ireland at Windsor Park, has told Portuguese newspaper O Jogo that he is happy to remain at the Bernabeu.
"There was the possibility of joining Manchester United, but I stayed at Real Madrid, and that's what matters," he declared.
"It would have been bad if I had to stay despite having the chance to move to a better club, but that didn't happen as I already am at the best club in the world.
"I want to do better than I did last season, although I was at a very high level already. I want to show everyone that I deserve to be at Real Madrid."