The Rossoblu had been keen to get rid of the forward, who signed a lucrative contract with the club upon joining from Fiorentina during the January transfer window
Goal.com revealed on Wednesday that the Rossoblu had offered the Italy international to Inter, who are looking to bolster their attacking options following the departure of Diego Forlan and the seemingly imminent exits of Luc Castaignos and Giampaolo Pazzini.
Genoa, meanwhile, were keen to remove Gilardino's annual salary of €1.5 million from their wage bill. However, Preziosi insisted that the striker will not be forced out of the Luigi Ferraris.
"[Club CEO Pietro] Lo Monaco has spoken clearly: unless a player expresses a desire to move, and if he remains in our plans, he will continue to be made available to [coach Luigi] De Canio," the Rossoblu chief told reporters at a press conference on Friday.
"Things are different now. So, if Gilardino wants to stay, he will not move from here."
Gilardino joined Genoa from Fiorentina during the January transfer window for a reported €8m, and still has four years to run on his lucrative contract with the Grifone.