Valencia has started 18 of United's 33 top-flight matches in a disappointing first campaign with Moyes at the helm.
United have relinquished their Premier League title and sit seventh in the table, with only a remote chance of qualifying for next season's UEFA Champions League.
Moyes' side were knocked out of this term's Champions League at the quarter-final stage by holders Bayern Munich last week, with a 4-2 aggregate defeat ending the Scot's hopes of landing a major trophy in his first season at the club.
There have been reports of unrest within the Old Trafford dressing room throughout the campaign and, according to the Sunday People, Valencia has now gone public with his discontent.
"I would like to have more minutes, I would like to have a more important role here," the British newspaper quotes the Ecuador international as saying.
"In fact, I would have liked to have more importance inside the team.
"But I have not been given the chance. I hope next season will be better for me as well as Man United."
