Arsenal are considering selling this key player to Serie A side – report
Former Gunners chief executive Ivan Gazidis is heading up Milan's transfer operations after moving to San Siro last December.
The Rossoneri have identified the Uruguay international as a key target as they attempt to build a squad capable of qualifying for the Champions League.
Milan are exploring a two-year loan deal worth around £7m which would also include an option-to-buy clause of £34m.
Torreira impressed in his debut campaign at the Emirates Stadium, quickly becoming a fans' favourite for his tough-tackling style.
However, the ex-Sampdoria man could be on his way back to Serie A this summer.