The Italy international was watched by United scouts earlier this season, but the Premier League leaders decided against making a bid during the January transfer window.
The 24-year-old was part of Cesare Prandelli's Euro 2012 squad last summer, and the Torino centre-back has been identified as one of the most highly-rated young defenders in Serie 'A'.
Arsenal scouts travelled to Italy to watch Ogbonna during Torino's match against Atalanta - a fixture the 11th-placed Torino won 2-1.
Reports earlier this month had suggested the Serie 'A' club were prepared to hold talks with Ogbonna over his future, as Premier League clubs circled.
A Sky Sports report had claimed United were monitoring Ogbonna's progress, with a view to possibly bringing him to Old Trafford as a long-term replacement for Rio Ferdinand.
But Arsenal have now joined the race for the young defender. The Gunners' defensive frailties were clear to see during their 3-1 home defeat to Bayern Munich in the Champions League last-16 first-leg on Tuesday evening.
Per Mertesacker and Laurent Koscienly occupied the centre-back spots for Arsene Wenger's side against the German giants, but struggled for much of the match to keep tabs on the talented Bayern front-line.
Ogbonna is a product of the Torino youth academy, and the 24-year-old has five caps for Italy.