The Sunday Mirror says United boss Sir Alex Ferguson had a £20million bid for Benzema accepted by Lyon last summer, only to be priced out of the market when Real offered £30m.
But the 22-year-old has been unable to hold down a regular place in Real’s star-studded team, with Gonzalo Higuain in sparkling scoring form.
Madrid are believed to be ready to allow Benzema to go out on loan next season to rediscover his form.
The Spanish league leaders are unlikely to sanction a move to any club that will rival them in the Champions League.
But United hope Real’s stance is weakening and that they may be able to agree a cut-price deal.