The Croatia international has been pushing for a move away from the Santiago Bernabéu due to a lack of game time at the club, and a summer switch to Stamford Bridge seems most likely before potentially returning to Madrid to succeed compatriot Luka Modrić in their first team.
The 24-year-old's move to west London could be completed as early as Wednesday, according to the Sun, who claim that the midfielder will undergo a medical with the club on August 8 - just 24 hours before the transfer window closes.
Kovačić will become Chelsea's second major signing of the summer following the arrival of Napoli midfielder Jorginho, although free agent Rob Green has also joined the club's ranks ahead of the new season.
It is claimed elsewhere by the Times that the club's goalkeeping target Kepa Arrizabalaga will also undergo a medical on Wednesday ahead of the Athletic Club star's permanent move, which is believed to be in the region of £71m.
The report continues by outlining four Chelsea players who could be axed following Kovačić's arrival at Stamford Bridge. It is claimed that at least one of Danny Drinkwater, Ross Barkley, Ruben Loftus-Cheek and Tiémoué Bakayoko will leave the club this summer.
All four players have already been linked with moves away, but at this stage in the transfer window it would appear that Bakayoko is close to leaving.
The Frenchman is supposedly the subject of discussion between the Blues and Italian giants AC Milan, with the Rossoneri eager to sign Bakayoko on loan for the season.