Both Premier League sides have suffered from their repsective striker-related issues of late, with Chelsea's Diego Costa struggling to replicate last season's form and City's Sergio Aguero continually plagued by injuries.
Benzema, who recently turned 28, is widely regarded as one of the world's finest strikers but finds himself at the centre of controversy at present.
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The France international is currently facing criminal charges after accusations were made against him in a sex tape scandal regarding international teammate Mathieu Valbuena.
Benzema denies any wrongdoing and has repeatedly stated his innocence, but now sources close to him are claiming he is fed up by the lack of support showed by those around him at Los Blancos and could be interested in a move this winter.
Having joined Real Madrid from French side Lyon in 2009 for a reported fee of £25 million, the striker has since made 296 appearances in all competitions, scoring an impressive 148 goals in the process.
And the Frenchman has been in blistering form so far this term, scoring 11 in 11 league games, including a hat-trick during Madrid's 10-2 annihalation of Rayo Vallecano last weekend.
For both Chelsea and City, Benzema's goalscoring exploits could prove vital, with the former in dire need of reinforcements going forward given their lowly position of 15th with -6 goal difference.
The Citizens' away record this season, meanwhile, has seen them win just three out of eight games, scoring a mere eight goals in the process. Aguero's persistent injuries and Wilfried Bony's struggle to establish himself have proved a direct cause of this.
The Premier League pair's troubles extend further than just the goals department and find themselves in the midst of manager-related speculation, with Manuel Pellegrini under pressure to succeed and Jose Mourinho recently relieved as Chelsea manager.
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