Speaking exclusively to the London Evening Standard, Kjaer placed Arsenal at the front of the queue while making clear they are far from alone in their interest.
"We know that Arsenal have been looking into Osorio, but we also know other clubs in the Premier League have done the same," Kjaer said. "I also think it is well known that Bournemouth looked into him historically. There are other clubs in the Premier League also looking into him."
His assessment of the player was emphatic.
"In my opinion, he serves the Premier League very, very good because he has a hard-working mentality.
"His defensive transition runs are amazing, his top speed is amazing. Left-footed right winger, can go to the right and to the left of his direct opponent. He has the full package, seen as the next Alexis Sanchez in Chile. I can see him making a difference in the Premier League."
Osorio joined Midtjylland from Universidad de Chile in the summer of 2023 for €5.2 million and has since made over 90 appearances for the Danish club, contributing eight goals and ten assists across all competitions last season.
Liverpool previously had a bid of under €10 million rejected, well below the club's valuation. Midtjylland have since extended Osorio's contract to July 2030, strengthening their negotiating position, and club president Claus Steinlein has said publicly he expects a record Danish transfer fee — a benchmark currently set around €24 million — when the winger eventually departs.
Kjaer also suggested Arsenal's first league title in two decades may have changed how they approach targets.
"Arsenal as the champions — I think that might also rock their boat a bit and see how we can keep this position for next season. I don't know. We will see where it all ends up."