Three names were recently said to be on the IFF's shortlist, which is topped by Chelsea boss Antonio Conte, while Luigi di Biagio also remains in contention after being promoted into the job on a caretaker basis.
Asked for an update on the search for a new manager, Costacurta told reporters a day after the 2-0 loss to Argentina: "We will tell you the name of the new coach on May 20. If we have the opportunity to get someone better than Di Biagio - and I don't think there are many - then we will.
"I think it's more important to think about the players we have, because I doubt Jose Mourinho or Pep Guardiola would have done better yesterday. Di Biagio is a coach who has improved year by year, climbing the ranks within the federation.
"It's very likely that he will either be the new coach or enter into the staff of the new coach. Gigi is very talented. I have been watching him closely and can confirm those within the federation see the situation the same way I do."