Benzema has endured a difficult season at Real Madrid so far.
The France international has scored 12 goals in 26 games but his displays against the bigger teams have left much to be desired.
He failed to finish off his chances as Madrid were dumped out of the Copa del Rey on Wednesday.
And Madrid president Florentino Perez is said to have grown tired of the striker’s erratic performances.
Real Madrid have reportedly offered Karim Benzema to Arsenal and Chelsea
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Karim Benzema has been at Real Madrid since 2009
It is claimed the Madrid chief believes Benzema’s best days are now behind him.
Cristiano Ronaldo is key to the deal with Madrid considering whether to use him as a natural centre-forward.
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The former Manchester United star is a winger by trade but has been utilised up front in the past.
Paris Saint-Germain are also keen on signing Benzema before Tuesday’s 11pm deadline.
However, his love for old club Lyon could scupper PSG’s chances of signing him.