Piatek leads the Serie A goalscoring charts this season with nine strikes in seven games, turning him into one of Europe's most sought-after young strikers.
Napoli, however, are looking to steal a march on their rivals, with De Laurentiis confirming he is negotiating a bid.
"I have spoke with both [Genoa president] Preziosi and his agent, now we have to meet," the Napoli chief told reporters on Monday.
"But I do not want to end up in a bidding war, with all due respect to the player, I must also see if he can perform at Napoli like he has done at Genoa."
Piatek started his career in Poland with Lechia Dzierzoniow and Zagłebie Lubin, before earning a move to Krakow side Cracovia in 2016.
A breakout season in 2017-18 saw him net 21 goals in 36 Ekstraklasa, a tally that convinced Genoa to take him to Italy at the start of the current season.
His stay at the club looks set to be a short one, however, with comparisons to Bayern Munich's Poland star Robert Lewandowski emerging due to his exploits in front of the net.
Piatek's talents have also been noticed in the Poland set-up, having been handed his debut for the national team in September against the Republic of Ireland.