The Sun says the Torino centre-back, 24, has shone over the past two seasons, helping his club win promotion back to Serie A.
Ogbonna, born in Italy to Nigerian parents, is under contract until 2016 and rated at around £16million.
The defender who can also play at left-back, was called up by Italy while still a Serie B player.
Rival bosses Alex Ferguson and David Moyes now plan to have Ogbonna watched in Torino’s clash with Inter Milan today — with views to a January move.
And a Torino club source said: “Ogbonna would not have any problems in adapting to English football.”