The Dutchman was cut loose by new manager Ruben Amorim on Monday after leading the team for three weeks as caretaker.
Andre Ooijer, the former Blackburn Rovers defender, is close to Van Nistelrooy, having been on his staff at PSV.
He told the Mirror: "Ruud wanted to stay as assistant. He would have said something if he wasn't interested in working with Amorim.
"Ruud has had the chance to be manager or head coach at other clubs, but he deliberately did not take those jobs because he wanted to work at the club he loves most.
"Don't forget Ruud did really well at United, a draw against Chelsea and victories in the other three matches. You could tell the fans liked him so much. You only had to watch how they reacted to him after his last match.
"He did so well under difficult circumstances. He has stayed unbeaten as a manager.
"If Amorim does not get the right results in the first few games, the crowd could start calling again for Ruud."