The Dutchman, a former United player who left the club in 2006 after a falling out with Sir Alex Ferguson, was the assistant to Erik ten Hag this season.
After Ten Hag lost his job on Monday, Van Nistelrooy was tasked with managing the team until a permanent successor is hired.
He stated this week: "At the time, I wasn't thinking that, to manage. I left in 2006, so a long time ago to Real Madrid and to perform and to play and to get the most out of yourself as a player.
“After that, you start thinking about coaching. But it was special to come back to this club and to this city that I liked and I loved so much and still do, and I enjoy it. I enjoy being around Manchester, I enjoy working with the players and staff here. There's not many that were there when I played, but there's a still a couple. I see familiar faces. More or less the same quantity of people from that time."
He added on his affinity with the club: "It's the other side of being a football player or a manager or an assistant. It's completely opposite sides. My team talk, I'm standing in front of the team telling them about what Manchester United is about and what it is to play at Old Trafford. Telling them what songs are being sung by the fans and why.
“So, you try to transmit that feeling, that lovely feeling playing for this club. It's a proud moment sharing that with players, and what they were transmitting on the pitch, and the way it interacted with the supporters. Also towards me, it was unbelievable support and it only strengthens me to do the utmost, especially short term, for Chelsea."