Assistant manager Phelan and goalkeeping coach Steele have been given their marching orders by incoming boss Moyes who is planning to use his own backroom staff rather than Sir Alex Ferguson's long-serving men.
"I would like to put on record the club's thanks for everything Mike and Eric have done," Moyes said in a statement released by United.
"I will be looking to appoint my own members of coaching staff at some point in the future.
"However, I'd like to take this opportunity to give them both our best wishes for their future careers."
Executive vice-chairman Ed Woodward added: "I'd like to thank Eric and Mick for the enormous contributions they have made to the success of Manchester United in their time here.
"Their professionalism, ability and diligence have been a credit to them. They are top-class football men and everyone here wishes them well for long and successful careers in the game."