The coach's contract expires at the end of the season.
Alessandro Moggi told Sportmediaset: "The most important thing is to underline how well Gilardino is doing.
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"A young coach with important game ideas. He won the Serie B championship last year and in his first season in Serie A he is proving to be very capable and will certainly be one of the most important coaches of the Italian future.
"It is normal to talk about the contract given that it expires on 30 June 2024 but today Gilardino's mind is totally dedicated to the end of Genoa's season."