Hodgson welcomed Moyes to Italy when he was in charge of Udinese to assist the then-Preston boss with his pro-licence.
Hodgson said: "David has been a friend since the time he visited me when I was at Udinese. He spent a week with me so we've been more than colleagues and acquaintances for a long time.
"I've followed his career very closely in the last five years, often on the bench on the other side from him - and I think he's done a fantastic job at Everton.
"He's a man whose record speaks for itself.
"Manchester United had an enormous choice when Sir Alex stepped down. They decided David had the qualities they were looking for."