Having spent the 2018-19 season on loan at Stamford Bridge, Kovacic could see that deal extended or be signed permanently by the Blues before July 1.
As Goal previously reported, Chelsea would be able to circumvent their impending transfer ban if they wish to keep Kovacic due to them already holding his registration from his time with the club last term.
The 25-year-old is yet to hear from the west London outfit, however, and despite having signalled that he would wish to stay in the English capital he has been left in limbo following the coaching upheaval within Chelsea.
Maurizio Sarri was keen to keep Kovacic at the club, but the Italian coach's departure to join Juventus means the futures of many players are less clear.
Derby County boss Frank Lampard is being lined up to replace Sarri and return to the club where he holds legendary status as a player, and it is unclear whether he would want to keep Kovacic within his squad.
Mason Mount, who played under Lampard for the Rams on loan last season, has secured a visa to join the Blues on their pre-season tour of Japan, and as such the number of places within the club's midfield has been further limited.
Madrid, meanwhile, are keen to raise funds having spent in the region of €300 million (£270m/$340m) on five players already this summer, including €100m on Chelsea talisman Eden Hazard.
They remain keen on signing Paul Pogba from Manchester United while a move for Paris Saint-Germain star Neymar is also on the table, and as such players must be sold to ensure the required finances are in place.
Kovacic is among 13 players who Zinedine Zidane deems surplus to requirements, with Gareth Bale, Isco, James Rodriguez and Keylor Navas also set to be moved on if Los Blancos can find buyers.
Inter and AC Milan have both been linked with Kovacic should he not remain at Chelsea, though it is understood that Sassuolo's Stefano Sensi is the midfielder being lined up by former Blues boss Antonio Conte at the former.
Officials from Milan, meanwhile, met with Madrid last week as they too plan a summer revamp following the departure of manager Gennaro Gattuso and sporting director Leonardo.
Kovacic is not the only loanee whose future remains unclear following a spell at Chelsea, with Gonzalo Higuain having disappointed during his time at Stamford Bridge.
With Sarri having taken over at the Argentina international's parent club Juve it has been expected that he would be given a second chance to make an impact at the Allianz Stadium.
The Italian champions have begun talks with his brother and agent, Nicholas, regarding his future, but it seems unlikely it will be at Chelsea.
Additional reporting from Romeo Agresti.