The young forward is one of the world's hottest properties at the moment, with Inter Milan also reported among the clubs considering a big money bid.
Speculation has increased that the Spanish giants are set to up their efforts to bring Benzema to the Bernabeu after failing in their bid to land Valencia striker David Villa.
Lyon, and the player himself, have always claimed he will remain at the Stade Gerland at least until next summer's World Cup and the club are confident that will happen.
"The only new fact is that Real Madrid got in contact with us a few days ago to know the position of the club on the subject of Karim Benzema," Lyon's director of communications Olivier Blanc told Marca.
"Benzema and Lyon confirmed their initial desire of him staying here. We know that the player wants to stay a year longer at Lyon and that we want to keep him."
Benzema came through the ranks at the French giants and burst onto the scene in the 2007-08 season with 31 goals in 51 games, a streak that saw him named Ligue 1 player of the year as well as top the league's scoring charts.
The last campaign was less impressive for both club and player, as Lyon ended the season without the Ligue 1 title for the first time in eight years, but the France forward nevertheless scored 23 goals in all competitions - an average of a goal every other game.
Can United compete with Real in the transfer market? Have your say.
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Red Dawn (25/06/2009 at 12:22)
He won't be very happy I'm sure, although he is likely to be evicted in time for pre-season training.
MUFC1878 (25/06/2009 at 12:45)
Was thinking exactly the same thing - Scary how much they look alike!
edinburgh red, edinburgh (25/06/2009 at 13:17)
MUFC1878 (25/06/2009 at 13:37)
If I was Helene Bonham Carter I would find your comment offensive and inappropriate!
It's definately Berbatov - you can tell from the glowing personality and hyper active drive she/he has
Bradley (25/06/2009 at 13:39)
salfordite, Salford (25/06/2009 at 14:18)
salfordite, Salford (25/06/2009 at 14:19)
boyfromburrenway, exile czech republic (25/06/2009 at 15:38)
good sign then if hes considering his finances,...maybe he needs a signing on fee........
paul leigh (25/06/2009 at 16:02)
dog hardy, manchester (25/06/2009 at 16:12)
Red Dawn (25/06/2009 at 17:08)
That is what I have been told, but it's best to just treat it like all other rumours at the moment.
He'd be a good signing though and provide us with a different option up front.
A Mancunian, Manchester (25/06/2009 at 18:01)
Red Army (25/06/2009 at 19:17)
The red divil, Angola (26/06/2009 at 00:28)