Meulensteen has claimed Moyes did not offer him a role after succeeding Sir Alex Ferguson last summer, a stance at odds with the United boss.
It appears now both men agreed to part ways and Moyes said: "I could turn it around and say Rene didn't keep any of the backroom staff when he went to Fulham. Rene got the opportunity to stay but I think the decision he took was right.
"He made the right decision and I believe I made the right decision as well. It was the right time for everybody. I knew it was going to be a big job and it has proved just that. I have the chance to build things up."