The terms of the move remain unclear but Madrid are keen to loan the 24-year-old without an option to buy, while Chelsea will push to include a purchase option as talks between the clubs continue.
Chelsea had tried to sign Alexsandr Golovin but they were snubbed as he chose AS Monaco in the final hours for development reasons.
The Blues are also likely to offload more than one senior midfielder before the transfer window closes on Thursday.
Meanwhile, the future of Chelsea goalkeeper Thibaut Courtois is keeping dialogue open between the clubs, as he looks set for a €35 million (£31m/$41m) move to Real Madrid.
The Chelsea No.1 is keen to be closer to his two daughters in the Spanish capital after his three-year loan spell at Atletico Madrid from Stamford Bridge.
Athletic's Kepa Arrizabalaga looks set to have his €80m (£71m/$92m) release clause activated by Chelsea, which will not only make him the club's record signing, but also the most expensive goalkeeper of all time.
Chelsea have informed the player and the Basque club of their intent as they aim to replace Courtois, who hasn't turned up for his first two days of training at Cobham, despite being due back on Monday.
Willy Caballero is expecting to go back to being the club's second goalkeeper, after starting in Chelsea's 2-0 Community Shield defeat to Manchester City.