Gazzetta dello Sport reported that the 53-year-old Italian has agreed a two-year deal worth two million euros ($2.3 million) a year which will be penned after the end of the Russian season which concludes next Sunday.
Zenit are fifth in the Russian Premier League, ten points behind already-crowned champions Lokomotiv Moscow with one match left.
The Russian club are reportedly reluctant to let Mancini leave and the Italian Football Federation FIGC do not want to pay any compensation to release him from his contract.
The former Italian international joined the Russian club last year on a three-year deal thought to be worth 4.5 million euros ($5.3 million) a year.
The annual budget for the future Italy coach and his backroom staff has been fixed at five million euros.
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 November.
Italy's under-21 coach Luigi Di Biagio has been filling the position on an interim basis.
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. He also won Italian Cups with Inter, Fiorentina and Lazio.
Meanwhile, Gazzetta also reported that Mancini wants to bring back Mario Balotelli into the national fold, and will ask goalkeeping legend Gianluigi Buffon to play against the Netherlands for a farewell appearance in Turin on June 4.
In-form Nice striker Balotelli, 27, has scored 13 goals in 33 appearances for Italy but has not been picked since the 2014 World Cup.
Former Bayern Munich coach Carlo Ancelotti turned down the opportunity to coach the four-time world champions saying he preferred to stay in club coaching.
"Mine was a choice, I want to coach a club," Ancelotti told Milan TV. "It was a difficult decision, but I have to listen to what I feel.
"Roberto Mancini is an excellent coach, he is motivated and has international experience. If the next coach was him, I think the national team would be in good hands."