Milik has been prolific for Dutch side Ajax this season, scoring 21 goals in 31 Eredivisie games.
Reigning Premier League champions Leicester are hoping to secure a deal for the 22-year-old, with The Mirror claiming boss Claudio Ranieri is holding talks with Ajax, with the striker valued at £15 million.
They are not the only Premier League side interested, with Spanish newspaper Estadio Deportivo claiming both Liverpool and Everton are keen, along with Spanish and Italian sides.
Leicester can offer Champions League football next season, and being able to win the race for Milik will be an early statement before the transfer window opens.
It is in their interests to get a deal wrapped up early with Milik set to feature at Euro 2016, potentially having the ability to increase his price tag and attract further suitors.