Madrid can't wait for Chelsea to find out if their transfer ban appeal has been successful and are moving ahead with plans to sell the Croatian midfielder, according to the Daily Mail.
Maurizio Sarri has expressed his admiration for Kovacic and wants him to stay beyond his current season-long loan.
"I want to try with Kovacic," said the Italian.
"He is only on loan, of course, but I'd like very much that Kovacic will stay with us.
"You know very well that, in that position, we have some problems because we have only Jorginho. In my opinion, Kovacic could become a very good central midfielder."
Kovacic is seen as dispensible by returning Real Madrid coach Zinedine Zidane, who is trying and clear room on the wage bill for players like Chelsea's own Eden Hazard.
The 24-year-old Kovacic could be joining Russia's league leaders in Zenit, who look likely to be back in the Champions League next season.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.