Vlahovic has been linked with a host of top European clubs after scoring 16 goals in 20 Serie A appearances this season.
Arsenal has been strongly linked with a move for the 21-year-old Serbia international, who is the leading scorer in the Italian top flight.
Fiorentina is reportedly holding out for a fee of €75million for their prized asset.
Barone has urged Vlahovic to make his intentions clear amid constant speculation over his future, which began even before the transfer window opened this month.
He said: "Dusan is talked about every day, but I would like to talk about the group. We expect an answer from him on the future, obviously we have our plans and we want to take action.
"Two days ago I had a conversation with Dusan, saying if you want to stay, please state it publicly and sign. A city cannot wait, but from what I have heard he does not seem willing to speak and therefore we as a club have to make decisions."
Barone said Fiorentina are willing to consider bids for Vlahovic, whose contract runs until the end of next season, but stressed they have not received any.
"All offers will be evaluated. But nothing has arrived at the moment." he added.
Vlahovic was named in the starting line-up for his side's Coppa Italia clash at Napoli on Friday (AEDT).