Kovacic left Inter for Madrid last year, but endured a difficult first season, making just eight La Liga starts.
The Croatia international failed to make Zinedine Zidane's 18-man squad for last Saturday's Champions League final against Atletico Madrid, and has been heavily linked with a move away from Spain.
Juventus and AC Milan are said to be interested in Kovacic's services, but Ausilio would like to see the 22-year-old spurn their advances in favour of trying to establish himself at the Santiago Bernabeu or a return to the Nerazzuri.
Ausilio told Radio Umbria: "If he has to come back to Italy, selfishly I hope it's not to AC Milan or Juve. Inter would hate to see Mateo with another shirt because we are fond of him.
"We always have an interest in a guy like Kovacic, but we cannot set anything. I hope he continues on his path at Madrid, a team that has just won the Champions League.
"It's true that he did not play much, but Mateo, for his age, has many appearances in professional leagues such as la Liga and Serie A.
"If he is patient, he may become important for Real."