The Sunday Mail reports that Mancini has been offered the deal as contract talks with City drag into another week.
Reports have long suggested that Mancini is close to signing a new contract with City.
Russia are desperate to bring in a new high-profile boss ahead of the World Cup.
Russian sports website Sports.ru claimed Mancini had agreed a £6million-a-year deal with the Russian Football Federation, but City quickly rebuffed the story.
Mancini has been offered a three-year deal worth £5million-a-year by City, but the Italian is keen on a longer deal.
City sources told the Sunday Mail: “We’re having our discussions on his contract that started at the end of the season.
“We think it’ll conclude in the coming days, get sorted and signed off. He’s definitely off to Austria on Sunday with the team, not Moscow!”
