The Scot has the unenviable task of succeeding Sir Alex Ferguson, whose 26-year reign at Old Trafford was laden with silverware. Moyes did not manage to win a trophy during 11 years at Everton - admittedly working with a fraction of the Red Devils' budget - but Van Nistelrooy said that would not be acceptable at his new club.
"Success at Manchester United? It doesn't matter which manager it is you need to compete on all fronts so that will be expected of him and I don't think that will be a surprise (to Moyes)," he said.
"I think everyone expects a league title, FA Cup, Champions League trophy and playing some good football. In that sense the pressure is full on but I think he knows what he is getting into. I don't know him personally but he looks like a guy who will handle that."
Moyes does not officially take charge until July 1 but has already been familiarising himself with the set-up at the club.
Recruitment of his backroom staff has yet to be finalised but assistant manager Mike Phelan and goalkeeping coach Eric Steele have already been moved on.
More are expected to follow - coach Rene Meulensteen is virtually certain to depart despite being offered a role in charge of the youth set-up - as remnants of the Ferguson era are gradually whittled away.
However, former United defender Henning Berg does not believe Moyes will have any issues putting his own stamp on things and gaining the respect of the players.
"I don't think (exerting his authority) will be a problem," said the Norwegian, who had a brief spell in charge at Blackburn last season.
"The way he leads I am 100 per cent sure he will have the respect and authority any way. I am sure he is looking forward to the job. Everyone says it will be very difficult after Sir Alex Ferguson - and it will be - but it is a great opportunity to be honest."