"Without the support of the fans there is, as we issue this note, a real risk that Heart of Midlothian could possibly play its last game on Saturday 17 November against St Mirren," the club said on their website.
"This isn't a bluff, this isn't scaremongering, this is reality."
Lithuanian businessman Vladimir Romanov is the principal shareholder at the Edinburgh-based club who were founded in 1874.
The Scottish Government said it was ready to offer help.
"The Scottish Government stands ready to assist in any way it can, including making contact with HMRC (Her Majesty's Revenue and Customs) if necessary," a spokesperson said in a statement.
"It is in everyone's interests to find a solution which ensures that Hearts can continue in business while also meeting their obligations to the tax authorities."
Hearts, fourth from bottom in the SPL with 13 points from 12 matches, were domestic champions in 1895, 1897, 1958 and 1960 and have also lifted the Scottish Cup eight times.
Rangers, Scottish champions a record 54 times and one of the best supported clubs in Britain, were demoted from the Premier League to the fourth tier this season after collapsing under crippling debts.
- Sports & Recreation