According to AS, Real will put forward terms that will see the French international's pay significantly jump from €5m per season to €8m, with an expiry date of 2021 or even 2022.
Currently, Benzema is contract to Real until June 2019. He has been linked with various moves in recent seasons, Arsenal being just one of the alleged suitors - albeit with little substance, but it looks as though he could remain with Los Blancos well past his 30th birthday.
Were Benzema to sign an extension, it would almost certainly have an impact on Real's summer transfer plans. Both Sergio Aguero and Pierre-Emerick Aubameyang have been linked with high profile moves to the Bernabeu, while it could serve to push Alvaro Morata out of the door.
The young Spaniard has continued to play second fiddle to Benzema as Real's central striking option since returning to the club. He has been heavily linked with Chelsea, with a long-term show of faith in the more established star perhaps enough to tell Morata he'd be better off elsewhere.
Benzema himself has lived in the shadows throughout his time in Spain. His 2009 arrival had followed Cristiano Ronaldo's world record move from Manchester United, while Gareth Bale also joined the club as another world record signing more recently.
Yet Benzema has maintained a high output in Real colours. He's scored at least 20 goals in every complete campaign since 2010, and enjoyed his best ever La Liga tally just last season with 24 strikes to his name.
To continue the discussion on Twitter follow @jamiespencer155