The 24-year-old defender has been heavily linked with a move to the Premier League giants, after impressing in Serie A over the past few years, after coming through the Italian club's youth ranks.
Red Devils boss Sir Alex Ferguson is keen to bring in a centre-back as a replacement for Rio Ferdinand or Nemanja Vidic, with both players approaching the latter stages of their careers.
Phil Jones and Chris Smalling are seen as Manchester United's future first-choice pairing, with Jonny Evans having also impressed in the Old Trafford first-team this season.
However, Ferguson could act in the summer and attempt to lure Ogbonna to England if talks break down over a new contract with Torino.
Urbano Cairo, the club's chairman, told Tuttosport that interest is mounting in the Italy international.
"A lot of clubs are still very interested in Ogbonna," he admitted. "In June we'll sit down and evaluate together what to do next. Maybe he'll have a desire to stay here for a bit longer."
At 34, Ferdinand is set for a one-year extension on his current contract that expires at the end of the season, while Vidic's future is uncertain as he continues to battle his troublesome injury problems, since damaging cruciate knee ligaments in 2011.