The 27-year-old forward wants to leave the Giallorossi outfit this summer and has been strongly linked with a move to Juventus in recent months.
"It's Vucinic that wants to leave," said Roma sporting director Walter Sabatini. "If it was up to us, he could stay.
"We have granted him [Vucinic] his wish to listen to eventual offers that arrive but Juve has not stepped forward so far.
"I can say that until tomorrow afternoon [Thursday], Vucinic's price will be the same as it is now, but after that he will cost more."
Vucinic, who is under contract with Roma until June 2013 and is valued at around €20million, had a strained relationship with the Giallorossi fans last term.
Roma supporters booed the former Lecce star after his disappointing campaign.
Vucinic scored ten goals in 28 appearances to help Roma finish sixth in Serie A.
Roma, meanwhile, are closing in on the signing of Espanyol goalkeeper Carlos Kameni.
With Alexander Doni and Julio Sergio set to leave the club this summer, Roma are looking for a new keeper.
Roma had held talks over signing Ajax's Holland international Maarten Stekelenburg but the deal failed to materialise.
When asked about Kameni, Roma sporting director Walter Sabatini told Sky Italia: "We are one step away from acquiring him."
The Cameroon international is under contract with the Spanish side for another two years.
The 27-year-old is valued at around €5million and has also been linked with Fulham this summer.