According to local media reports, Montella is among the front-runners to replace Milan coach Clarence Seedorf should the Dutchman be sacked in the summer as widely expected.
"There is no chance that Montella won't continue to be Fiorentina manager. He is staying 100 percent," Cognigni said during a meeting with reporters in Florence on Friday evening.
"Montella has not once showed us any signs that he would like to leave. He has no desire to abandon his work."
Former AS Roma striker Montella guided Fiorentina to a fourth-place finish and the final of the Italian Cup this season despite missing star attackers Giuseppe Rossi and Mario Gomez for large duration of the campaign.
His side host Torino in the final weekend of the Serie A season, having already confirmed their place in the Europa League for next season.
Montella, reportedly, wants assurances that the club will match his ambition to challenge for the top honours.
(Reporting by Terry Daley; editing by Sudipto Ganguly)
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