"Yesterday [Monday] in Rome there was a meeting ... with Roberto Mancini, who said he was ready to end his relationship with Zenit and coach the national team," Fabbricini said in quotes carried by Gazzetta Dello Sport.
"We agreed that we'll start to talk about figures and details on May 13th, at the end of the Russian championship.I want to state that nothing is decided. We've told Mancini clearly that the FIGC has a budget for the role which it does not intend to deviate from."
The annual budget for the future coach and his backroom staff has been fixed at €5 million.
Mancini, 53, met with the Italian FA sub-commissioner Alessandro Costacurta and team manager Gabriele Oriali for over two hours in Rome on Monday evening.
It was reported over the weekend that Carlo Ancelotti had turned down the job, with Gazzetta reporting that the federation had offered the former Manchester City manager a two-year contract until 2020 worth €4 million a year.
"Coaching the national team would be prestigious and a source of pride for me because Italy are one of the most important teams in the world," Mancini told Italian radio on Monday.
During his 17-year coaching career Mancini led Manchester City to their first English league title in 44 years in 2012, and won three Serie A titles with Inter Milan and Italian Cups with Inter, Fiorentina and Lazio.
The former Lazio and Sampdoria forward never became a regular with Italy during his ten-year international career, during which he won 36 caps and scored four goals.
Italy have been without a permanent coach since Gian Piero Ventura was sacked after they failed to qualify for the World Cup for the first time in 60 years after losing to Sweden in a two-legged playoff in November.
Italy's under-21 coach Luigi Di Biagio has been filling the position on an interim basis.
Photo: Tiziana Fabi/AFP