The report claims the France international is now more open to a Arsenal move having snubbed the chance to speak to the Gunners over the summer.
Arsene Wenger is thought to have prioritised the signing of a top-class striker in the winter window, as he looks to provide support for Olivier Giroud.
Arsenal have asked to be kept up to date with the latest regarding Benzema's availability, and with the Frenchman struggling for consistency at the Bernabeu, he could be sold to the highest bidder in January.
Arsenal have plenty of cash - reports placed their budget over the summer at £70m - and the club enjoys a solid relationship with Real Madrid having signed Mesut Özil from the Spanish capital in August.
Benzema's contract expires in the summer of 2015, with Arsenal thought to be willing to spend around £22m on the forward.
Despite a recent upturn in form - he has four goals in his last four games - there's a feeling he could be surplus to requirements at the Bernabeu, especially following the £86m outlay on Gareth Bale.
Wenger is a long-term admirer of the France international - who has been replaced by Arsenal forward Giroud in the national team starting lineup - but much could depend on Real Madrid's ability to sign a replacement.
Their first-choice target is thought to be Liverpool's Luis Suarez, but there is an expectation that the Uruguayan will not move until the summer at the earliest with the Anfield club in such a strong Premier League position.
That means Arsenal may be forced to wait until next summer, although Wenger is keen to strengthen his attacking options at the earliest opportunity.
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