The Zenit St Petersburg manager met with the Italian FA sub-commissioner Alessandro Costacurta and team director Gabriele Oriali in Rome on Monday evening.
Mancini, 53, has always been reported as in the running for the Azzurri job, but Chelsea manager Antonio Conte and Carlo Ancelotti were also on the shortlist, according to La Stampa (h/t Football Italia).
Gazzetta added that the former Manchester City boss was offered a two-year contract until 2020 worth €4 million (S$6.4million) a year.
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Roberto Mancini had previously denied landing the Italian manager's job when he talked to Italian Radio on Monday.
Italy have been managed by Italy U-21 manager Luigi Di Biagio as a caretaker since the sacking of Gian Piero Ventura.
Mancini led City to their first Premier League title in 2012.
If Mancini does take the job, it could signal a return to the national team for Mario Balotelli.
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Zenit however are not happy that Mancini is leaving having only taken over last June, but the club are only fifth in the Russian Super League standings with two matches remaining.
Nantes coach Claudio Ranieri had been a secondary candidate.
Mancini has managed Fiorentina, Lazio, Inter Milan (twice), Man City and Galatasaray in a successful managerial career.
The 53-year-old former Italian worldwide is confident however he can reach a deal with the Russians.
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Brown is entering the final year of his contract and San Francisco drafted McGlinchey ninth overall to take his role. The 35-year-old's 68 touchdown receptions only trails Dez Bryant (73) and Bob Hayes (71) in team history.