Former AC Milan defender Costacurta has been given the job of finding a replacement for Gian Piero Ventura, who was sacked after Italy's shock failure to qualify for the World Cup last November.
And with under-21 boss Luigi Di Biagio in charge on an interim basis, the four-time world champions suffered defeat in their first game since, losing 2-0 to Argentina in Manchester on Friday.
"We will tell you the name of the new coach on May 20th," said Costacurta, who told a press conference that Di Biagio remained among the potential candidates.
"If we have the opportunity to get someone better than Di Biagio, and I don't think there are many, then we will."
Among the names being touted are Zenit St Petersburg coach Roberto Mancini, Chelsea boss Antonio Conte, former Leicester coach Claudio Ranieri, now in charge of French club Nantes, and Carlo Ancelotti, who is available after being sacked by Bayern Munich.
"It's not easy to find someone able to do a perfect job, and we need to see what we have to do to build a team capable of qualifying for Euro 2020," said Costacurta.
"I think it's more important to think about the players we have, because I doubt Jose Mourinho or Pep Guardiola would've done better yesterday."
He added: "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've been watching him closely and can confirm those within the Federation see the situation the same way I do."
Italy play England at Wembley on Tuesday and have June friendlies lined up against France and the Netherlands.