The Rossoneri have reportedly submitted a bid that incorporates a two-year loan deal with a subsequent obligation to buy the Uruguayan midfielder.
Milan propose to pay a £7m loan fee followed by £34m to be paid in one instalment 24 months down the line.
Torreira earned plaudits in his first season at Arsenal after arriving from Sampdoria last summer for £26.4m.
But the tenacious midfielder recently revealed he was struggling to settle in London.
“I don’t know if there are many things that I enjoy,” he admitted to Ovacion when asked about living in England.
“I think it was better in Italy. England is a totally different world, a very large country.”
Gunners boss Unai Emery has only a reported £45m to spend on strengthening his squad this summer.
But Arsenal fans would be far from happy to lose Torreira, even if the money was to be reinvested in their squad.
And fans are pinning the blame for the Torreria situation on their club’s former CEO Ivan Gazidis.
Gazidis quit Arsenal to take up a role at AC Milan and is plotting to raid his old club just eight months later.
On hearing the news of Milan’s offer, one fan tweeted: “That’s a Gazidis bid if ever I’ve seen one.”
“Who does Gazidis think he is?” asked another.
“This disgraceful offer has Baldzidis written all over it,” added a third Gooner.
“Gazidis smoking the same thing when he bid for Suarez…” laughed a fourth.
A fifth supporter wrote: “Wow so this is what it feels like to be a club arsenal are looking to buy from then…”
“After seeing this I am 100% convinced Gazidis was behind that extra £1 for Suarez bid,” added a sixth.
“Its not a coincidence that AC have started this nonsense now that he is there.”