According to AS, Inter watched their former star during Tuesday's UEFA Champions League clash with Paris Saint-Germain, while Italian rivals Milan and Roma are also monitoring his situation in the Spanish capital.
He was quizzed after the match in Paris about the possibility of a return to Italy and his role with Los Blancos.
"I don't know, it's difficult to talk about going back to Milan right now. I loved Italy and Milan. You never know," he said to Premium Sport (h/t AS). "It's tough being on the bench, but I'm ready for every game. Today I played, and we won."
Kovacic started the match ahead of Luka Modric and Toni Kroos, who had spent time on the sidelines injured in recent weeks, and SB Nation's Lucas Navarrete hailed his showing:
The latter pair usually wrap up the starting places in midfield alongside Casemiro, so Kovacic is typically a back-up at the Santiago Bernabeu.
He may not quite be at their level, but he's an excellent player in his own right.
The 23-year-old's distribution is metronomic, and he excels at beating opponents in the middle.
OptaJose and WhoScored.com illustrated his skills:
Managing Madrid's Kiyan Sobhani noted how strong his game can be when he's able to tie his strengths together, as he did in the 3-1 win over Leganes last month:
According to Squawka, he has also won 2.88 tackles per 90 minutes in La Liga this season, so he can also count ball-winning in his impressive arsenal.
He has the potential to get even better as he gets older, so he'd make an excellent signing for his old side, Roma or Milan.
It seems he'd consider it, given his evident affection for his time in Italy, though with Modric now 32 his opportunities at the Bernabeu may increase.
If he begins to play a greater role, he'll likely be content to stay put.