The Viola are looking for a new coach after sacking Delio Rossi earlier this month.
Montella, who is under contract with Catania for a further season, has been strongly linked with the Viola hotseat and Cognigni told Gazzetta dello Sport: "Montella could be the right coach to start a new project.
"However, we know Montella is under contract with Catania and we know there are rules that have to be respected.
"We are monitoring the situation considering that Montella has expressed his wish to start a new adventure."
Montella joined Catania last summer and led them to a respectable 11th-placed finish in his first season.
The 37-year-old, who has also linked with a return to Roma this summer, is set to meet Catania executives later this week to discuss his future.