The 22-year-old winger, who has been described as the next Alexis Sanchez, has recently been linked with a move to the Emirates Stadium as Mikel Arteta looks to bolster his frontline.
Osorio, whose contract at the MCH Arena expires in June 2030, featured 49 times across all competitions for the Danish outfit last season, scoring eight goals and providing 10 assists.
Playing a key role in Mike Tullberg's side, the Chile international helped Midtjylland win the Danish Cup.
His impressive performances didn't go under the radar, reportedly attracting interest from some top European clubs, including Arsenal.
Last month, speaking to Chilean outlet ADN Deportes, Midtjylland's sporting director confirmed that Arsenal have been keeping tabs on Osorio.
"We know Arsenal have been interested in Osorio, but we also know other Premier League clubs have been the same," said Kristian Kjaer, who then explained why the Chilean could be ready to test himself in the English top flight.
"In my opinion, the Premier League suits him very well because he has a very hard-working mentality," continued the former Brentford scout.
"His defensive transitions are incredible, his top speed is incredible. He's a left-footed right winger who can go to either side of his direct opponent. He has everything. In Chile, he's considered the next Alexis Sanchez. I think he can make a difference in the Premier League."
According to ADN Deportes, Benfica have since joined the race to sign Osorio.
The report claims that Marco Silva was an admirer of the Chilean during his time at Fulham and now wants to bring him to Lisbon.
It adds that Osorio could be available for €12million, which is £10.3million, to €15million, which is £12.9million, but Midtjylland are yet to set an official price tag.
As it stands, it remains unclear whether Arsenal will lodge a bid for the forward, but Arteta is certainly keen to bolster his frontline this summer.