Gilardino set for China move
The 31-year-old could reunite with former Italy coach Marcello Lippi in China as Genoa look to reduce their wage bill.
Preziosi is unsure if Gilardino, a 57-time Italy international, will accept the offer, but he wants him to.
"We don't know whether Gilardino will accept or not," he told Telenord on Wednesday.
"If he doesn't, then we will keep him and continue to give him a great deal of money. If he doesn't, then we already have some alternatives.
"We want to bring the average age of the squad down and reinvest the money for a younger forward who we believe could be as strong as Gilardino."
Gilardino scored 12 goals in 26 league games for the Serie A side last season having spent the previous campaign on loan at Bologna.