The ex-Dutch international joined Hamburg SV last January after netting 46 goals in 68 games over three-and-a-half seasons with the Spanish giants but his contract in Germany runs out at the end of this season.
Real say they are in the market for a striker in the January transfer window as cover for Argentina's Gonzalo Higuain, who is out for four months after back surgery.
"There are lots of names being mentioned," Mourinho told a news conference at the club's Valdebebas training ground when asked about former Manchester United marksman Van Nistelrooy, 34.
"If in the end it's Ruud I'll be very happy," added the ex-Chelsea coach. "We coincided in England and I know him well as a player and person.
"It would be very good for us to have another attacker because we don't have many and we have a lot of matches to play."
HAMBURG WINNER
Van Nistelrooy said after scoring the winner in Hamburg's 1-0 victory at Schalke 04 on Saturday he would have to mull over his next move.
"It's not easy for me as it's not just any club that is coming calling," he was quoted as telling German media.
"Hopefully things will be sorted out soon. I can't leave the lads in the lurch by having my head elsewhere. Real Madrid is the only club that could change my mind."
Schalke coach Armin Veh told Sky Television no official offer had yet been made for Van Nistelrooy.
A central figure in Real's title-winning seasons in 2007 and 2008 and popular with fans, the Dutch striker left following the arrival of Karim Benzema and Cristiano Ronaldo.
Higuain's injury and Benzema's indifferent form prompted Mourinho to ask the club for an extra striker as Real seek to end Barcelona's two-year run as La Liga champions plus progress past the Champions League first knockout round.
Van Nistelrooy has scored 11 goals in 26 league appearances for Hamburg.
