The Azzurri are prepared to sell the Nigeria international at a starting price of 150m.
Il Corriere dello Sport says PSG are favourites in the hunt for Osimhen.
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The French giants are said to have offered 90m plus 40m in bonuses.
But PSG may get unexpected competition from English giants Liverpoo.
Osimhen could replace Darwin Nez, who has had major problems with goalscoring, or Mohamed Salah - who is still being chased by Saudi clubs.