The Nigerian has become one of the most sought after forwards in world football after 21 goals in 44 league appearances for Napoli.
Gunners boss Mikel Arteta is targeting a big striker signing this summer after the departure of Pierre-Emerick Aubameyang.
Alexandre Lacazette and Eddie Nkeitah are also entering the final months of their respective contracts.
However, their pursuit of Osimhen will depend on whether they have enough to cash to meet Napoli's asking price of £84million.
The north London club are currently in fourth on the Premier League and securing Champions League football could go a long way to signing Osimhen next summer.