The former Yugoslavia winger, who was Red Star's technical director when they won Europe's premier club competition in 1991, sent an open letter to Serbia's sports minister Alisa Maric asking her to allocate a loan in order to save "a fallen giant from going under."
"We are not asking for subsidies," Dzajic said.
"What we want is a short-term loan to keep Red Star afloat...which we would repay straight into the country's budget as soon as we can. Otherwise, Red Star may cease to exist," he added.
Dzajic's dramatic appeal came only a day after Partizan Belgrade basketball club president Predrag Danilovic, a key player as they won the 1992 Euroleague title, also asked the government to help Serbia's top side overcome a financial crisis.
Danilovic said Partizan's account had been frozen as the club owed 1.5 million euros of unpaid wages to players and another 1.2 million in unpaid taxes.
Red Star and Partizan were both heavily subsidised by the government in former communist Yugoslavia but were left largely to their own devices after the country's bloody break-up which went hand in hand with Serbia's economic depression.
Partizan, who won a record 12th successive basketball league title earlier this month, had somehow managed to stay afloat thanks to playing in the Euroleague, the continent's flagship club competition, and nurturing a production belt of talent regularly snapped up by wealthier rivals.
Red Star, however, have been unable to produce in the last decade players remotely as good as former captains Nemanja Vidic and Dejan Stankovic, hence becoming a debt-ridden liability unable to pay even the most basic bills.
Last week, their Portuguese coach Ricardo Sa Pinto quit after the board failed to make any major signings, making way for Slovenian Slavisa Stojanovic who had said he hoped to win Red Star's first league title since 2007 next season.
"With 25 league titles, 24 national cups, one European Cup and a World Club Cup trophy in our cabinet, Red Star is one of Serbia's biggest brands and a name that still merits respect across the world," Dzajic said.