The Italian guided the Neapolitan side to an impressive second place finish - amassing 91 points - while also playing some of the most exciting football on the continent.
Napoli have twice finished as runners-up to Juventus in the past three seasons.
Palmeri added that Ancelotti won't be announced until June 1 as that's the day after his Bayern Munich contract expires. He is the only coach in history to win the Championships of Italy, England, France and Germany.
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The 59-year-old joined Napoli three years ago and had been under contract until 2020.
The former Chelsea and Real Madrid boss had been heavily linked to the Italian national team but that role went to Roberto Mancini in the end.
According to Sky Sports Italia, Zenit have invited the 59-year-old former banker to Russian Federation to see Saint Petersburg and their training facilities.
It is claimed Sarri would also tell Chelsea to move for Senegal star Kalidou Koulibaly, who was strongly linked with a move to Stamford Bridge in 2016.
Ancelotti excels in being a father-figure to his players, offering a calming influence from the sidelines and, as was the case in Italy and Spain, being hugely popular with players and fans alike. "Serie A regains one of the excellences of our football".