Montella has been linked with a move to Juventus and Milan as the Serie A season comes to a close but Cognigni told reporters on Friday that the former Roma striker would '100 per cent' remain in Florence for the 2014-15 campaign.
Fiorentina are set to play Torino on Sunday in their final Serie A game of the season and Montella's men will finish fourth, regardless of the result.
"There is no chance that Montella won't continue to be Fiorentina coach," Cognigni said.
"He is staying, 100 per cent."
When Montella started as Fiorentina's coach in June 2012, the Florence-based club had just finished 15th in Italy's top tier but the 39-year-old has turned their fortunes around.
Montella has a winning percentage of 57.5 with Fiorentina and has led the club to consecutive fourth-placed finishes in Serie A, plus the final of this season's Coppa Italia, which they lost to Napoli.
Cognigni added: "There is no chance for him to leave and on his part there is no willingness to abandon the work begun in recent years."
La Gazzetta dello Sport also reported that Cognigni had claimed Borja Valero and Gonzalo Rodriguez had extended their contracts until 2018 and 2017 respectively.
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