Moyes was given the unenviable task of replacing the great Sir Alex Ferguson at the helm for the 'Red Devils', signing a six-year contract in 2013.
United had won the title the previous season, but according to Carrick, Moyes was confident that he could help the players improve regardless of their recent success.
Writing in his new autobiography, Between The Lines, Carrick said that Moyes was keen to improve the work rate of United's stars, something that didn't go down particularly well with the players themselves.
"I got on fine with David. Never had a bad word with him, never fell out with him. He arrived full of energy and ideas for how to improve us," Carrick revealed.
"I remember at one of his first meetings, David said to us, 'I believe I can make you better. I know you won the league last year and I've obviously been watching you. I can make you run more.'
"Being fair to David, he was looking for a chink, like what needed improving and where we were weak, so that he could take us to another level.
"I just think what he said didn't come across the right way."
Moyes would end up being sacked in April 2014, with United finishing the season in 7th place in the Premier League, though his quarter-final finish in the Champions League was better than successors Louis Van Gall and Jose Mourinho have managed.