The Russian billionaire is believed to be content with the job Benitez is doing at Stamford Bridge this season, but the report claims the Spaniard has no chance of keeping his post past the end of the season.
And Moyes is the man Abramovich wants to replace him. The Toffees boss was linked with the Blues job earlier this month, but it now seems he's top of the Chelsea owner's shortlist to manage the club.
The pressure on Benitez mounted following the Blues' 2-2 FA Cup draw with League One side Brentford, but although the former-Liverpool boss is unpopular among a section of Chelsea fans, the club is thought to be willing to stick with him until the summer.
And Gary Cahill backed the under-fire boss following the Brentford result, taking the positives from an otherwise thoroughly disappointing match.
The England international highlighted Chelsea's fighting spirit as evidence of player support for Benitez, as the west London outfit twice battled back to hold the Bees to a 2-2 draw.
When asked if Chelsea's second half display was a show of support for Benitez, Cahill said: "Yes, of course. There is a lot of pressure from the fans but we needed to just try to do the job.
"In the end, we grafted it out on a horrible day and we now need to show our quality at home. It was the character we needed to dig in and just get the result we needed.
"Obviously we wanted the win, but at least we have taken it back to the Bridge and hopefully we can finish the job there."
Still, a 'post-Rafa strategy' is being drawn up at Stamford Bridge, with the Chelsea heirarchy eyeing Moyes as the top candidate to replace Benitez.
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