Karim Benzema
Chelsea may challenge Arsenal to Benzema's signature. (Reuters)

Jose Mourinho will challenge Arsenal for the signature of Real Madrid's under-fire striker Karim Benzema when the January transfer window opens.

Chelsea's current striking options so far have failed to be a consistent goal threat or show that they are potent enough to lead the Blues' vibrant attack, with the quartet of Samuel Eto'o, Andre Schurrle, Fernando Torres and Demba Ba failing to register a single Premier League goal between them.

Portuguese publication A Bola suggest the Chelsea boss will ask Roman Abramovich to sanction a move for Benzema whose relationship with Real Madrid fans is deteriorating by the game, as the France international struggles to find his own form during Real's subdued start to the season.

The former Lyon star has failed to appease the partisan Real support in the club's opening seven games of the season and was loudly jeered by his own fans as his side fell to a 1-0 defeat to city rivals Atletico Madrid at the Bernabeu.

Carlo Ancelotti, whose start at the club is also being scrutinised, is coming under increased pressure from fans and the Spanish press to consider the options of Spain Under-21 star Alvaro Morata and academy starlet Jesse over the misfiring Benzema.

The Frenchman was heavily linked with a move to Arsenal during the summer after the club failed to secure deals for his former Real Madrid teammate Gonzalo Higuain and Liverpool's Luis Suarez.

The Gunners may be tempted to launch another effort in January after chairman Chips Keswick assured Arsene Wenger that he would have funds available to ensure the club's terrific start to the season can continue until May.

But given Chelsea's dearth of potent options, Mourinho may choose to pursue the striker he spent the last three seasons of his career working with, prior to his return to Stamford Bridge earlier this summer.

The 25 year old enjoyed his most prolific season under the Special One's tenure in 2011/12, striking 32 goals in 52 games to help Real Madrid reclaim the La Liga crown from Barcelona.

But after a season where Mourinho's troubles with the club were often mirrored by the performances of his side, the Frenchman failed to replicate such form with his role in the Los Blancos squad now under severe scrutiny.