The Times says Real are thought to have identified Heinze as a replacement for Roberto Carlos, who has agreed to move to Fenerbahce at the end of the season.
Barcelona, Juventus and Paris Saint-Germain, from whom United bought Heinze for £6.9 million in June 2004, are also understood to be in the hunt for the Argentina defender, who could buy out the remaining two years of his contract at Old Trafford after coming to accept that he no longer figures prominently in Sir Alex Ferguson's plans.