The Sun's Anthony Chapman reported Blues manager Antonio Conte "has decided to cash in on Costa at the end of the season as he looks to take the team in a new direction."
Chelsea believe they can sell Costa for a hefty sum, and the striker is ready to step into Benzema's role at the Santiago Bernabeu.
The Spain striker has accrued 18 goals and five assists in 30 appearances in all competitions this season.
As Bleacher Report UK's Garry Hayes noted, Costa has been among Chelsea's top performers:
Garry Hayes @garryhayes
Who is Chelsea's most indispensable player this season? #CFC https://t.co/bAXVZquepR3/7/2017, 1:53:05 PM
The former Atletico Madrid hitman has been revitalised under Conte, and while he remains an aggressive player, he has become somewhat more disciplined since the Italian took over.
According to Squawka, the forward has committed fewer fouls per game than in his previous two seasons at the club and received fewer bookings, too.
Bleacher Report UK's Sam Tighe and The Times' Alex Kay-Jelski remarked on his consistency of performance this year:
Sam Tighe @stighefootball
Diego Costa has made 25 Premier League appearances this season. He’s scored in 16 of them. Consistently making the difference for #CFC.3/7/2017, 9:17:58 AM
ALEX KAY-JELSKI @AlexKJTimes
Diego Costa has scored 18 goals this season but has only scored more than once in one match - away at Swansea. Consistent yet bizarre #CFC3/6/2017, 9:29:38 PM
Tighe also noted the striker can be a real handful for defenders even when he's not finding the back of the net:
Sam Tighe @stighefootball
@GaffneyVLC he had a bit of a rough patch where he didn’t score, so he just resorted to making a right nuisance of himself. Worked well.3/7/2017, 9:21:45 AM
Real aren't desperately in need of more firepower thanks to having centre-forward options in Benzema and Alvaro Morata—not to mention Cristiano Ronaldo and Gareth Bale alongside them—but Costa is of similar quality to Benzema and a year younger than the Frenchman.
Reinforcing at centre-back could be a more pressing need for Zinedine Zidane this summer, particularly if Pepe leaves. According to Manu Sainz of AS, Real will target Porto defender Felipe Monteiro as his replacement.
Indeed, the club have held talks with his agent Giuliano Bertolucci and "have reached a verbal agreement" that the defender will join in the summer.
Here's a look at the pacey and combative centre-back's highlights this season:
Pepe is 34, so replacing him with some fresh blood to compete with Raphael Varane and Sergio Ramos would be a wise move in the summer, and the 27-year-old could make a strong candidate.
According to Sainz, he has a release clause of €50 million but "sources close to the Portuguese club" believe a bid of around half that could suffice.