"The manager met chairman Bill Kenwright early yesterday evening and confirmed his desire to join Manchester United," Everton said in a statement on their website.
"The chairman, on behalf of the club, would like to place on record his thanks to David for the massive contribution he has made to Everton since his arrival in March 2002.
"He has been an outstanding manager."
Moyes will remain in charge for Everton's final two Premier League games of the season against West Ham United and Chelsea.
United have not confirmed whether Moyes will be their first new manager for 27 years.