Having already signed Lucas Torreira and Mattéo Guendouzi, Emery is reportedly keen on adding Kovacic to his squad to provide energy and guile to his midfield, as reported by the Express.
Kovacic, 24, has had a tough few years at Real Madrid after signing in 2015, failing to nail down a starting position, with competition from Toni Kroos, Casemiro and Croatia teammate Luka Modric.
The Real Madrid man will take a break after losing in the World Cup final to France on Sunday but has expressed his intentions to leave the Spanish capital.
Even the appointment of former Spain manager Julen Lopetegui has not made Kovacic contemplate staying. When asked about the possibility of staying at Real Madrid, Kovacic said: "We'll see."
Arsenal will face stiff competition from Roma and Manchester City if they want to secure Kovacic's services for next season.
Emery could be looking to fill an Aaron Ramsey-shaped hole in his squad with rumours of his exit from Arsenal refusing to go away.
Ramsey, whose contract is up in 2019, has been linked with moves away from the Emirates if he doesn't sign a new contract.
However, Emery has insisted that Ramsey has a key role to play in his plans and wants the Welshman to stay.
Emery has made five signings in total during his first transfer window as Arsenal manager, with Stephan Lichtsteiner, Bernd Leno and Sokratis Papastathopoulos joining Torreria and Guendouzi at the club.