Goal.com quotes Italian source Tuttosport who say that the Russian side also want to sign the Torino centre-back, who is rated at €20m.
The Cassino-born defender came through the Torino academy and has made 126 league appearances for the Serie A side.
His impressive displays led to a call-up to the Italy under-21 side, before senior team manager Cesare Prandelli handed him his first full Italy caps in 2011 - and he was included in Italy's Euro 2012 squad.
The 24-year-old centre-back has made four appearances for the Italian national side, and was linked with a move to Manchester United last month. United have troubles in defence - they conceded three at home to Tottenham, and another early goal away to Cluj in the Champions League this week.
And Manchester United have injury troubles at the back - Jonny Evans limped off against Cluj - while Nemanja Vidic has been ruled out for another two months.
In contrast, Manchester City have a number of options in defence, and signed teenage Serbia star Matija Nastasic from Serie A side Fiorentina during the transfer window.
But Manchester City boss Roberto Mancini is reportedly still interested in Ogbonna, who is higly-rated in Italy.
But it seems big-spending Anzhi have entered the Manchester battle to sign the tough-tackling Serie A star. Under the guidance of sporting director Roberto Carlos, Anzhi have been rumoured to be interested in several of Europe's big names.
The ambitious Russian side, who sit top of the Russian Premier League, made a splash in the European market when they signed Inter Milan star Samuel Eto'o and former-Chelsea defender Yuri Zhirkov.
Another ex-Chelsea employee, Guus Hiddink, manages the Dagestan-based team, who are looking for their first ever Premier League title and maiden qualification for the Champions League.