Real have been linked with a range of strikers - including Manchester City's Emmanuel Adebayor - following a long-term injury suffered by Gonzalo Higuain that has left the club with only Karim Benzema as a recognised striker.
Van Nistelrooy, 34, is another name believed to be under consideration, despite only leaving the Bernabeu to join Hamburg 12 months ago.
The Dutchman scored 64 goals in 96 games for Real in all competitions between 2006 and January 2010, winning back-to-back titles in 2007 and 2008, and Mourinho would be keen to see him return to the Spanish capital.
"There are lots of names being mentioned," Mourinho said. "If in the end it's Ruud I'll be very happy.
"We coincided in England and I know him well as a player and a person. I'd be very happy if it's him. It would be very good for us to have another attacker because we are not many and we have a lot of matches to play.
"At the moment he is a Hamburg player and we need to respect that situation. I don't want to talk of Van Nistelrooy. For you (the media) it's easy. I have to respect Hamburg."
The player himself claimed he would think about the move.
"I need a good weekend to mull it over,'' he said. ''It is also not easy for me, it is not any old team who is coming in for me."
HSV sporting director Bastian Reinhardt, however, told Sky that he expects Van Nistelrooy to honour his contract, "There is only one trend: he stays!
"Of course it is always an honour for a player when a request from Real Madrid comes. But we didn't get any offer yet and also the timing, being so close to the start of the second half of the season, is not perfect, as you can imagine.
"The player knows that there is an request, but the player also knows that he has a contract."