The 24-year-old, who has won five caps for Italy, has established himself as one of the country's best defenders since breaking into the Granata side in 2009.
The Gunners were said to have sent scouts to watch Ogbonna, but faced strong competition from long-term admirers Manchester United.
Torino chairman Urbano Cairo has already accepted that he could lose Ogbonna in the summer as a number of Europe's top clubs circle the water.
According to talkSPORT, Arsene Wenger will try to preempt bids from Serie A teams such as Juventus, AC Milan, Fiorentina and Napoli.
Real Madrid and Paris Saint-Germain are also thought to be interested.