The Edinburgh club, where Lithuanian Vladimir Romanov is the main shareholder, made an appeal last week for emergency financial help from fans.
Hearts, who have total debts of £24 million, face a winding-up order over a £450,000 tax bill and the deadline is Thursday evening.
They are open to allowing fans the chance to take ultimate control of the club but gave short shrift to an approach from a consortium by a group fronted by businessman Alex Mackie.
"At best it is opportunistic, an attempt to exploit what is a difficult financial situation at the club," Hearts said on their website.
"While the club remains open to genuine offers it will continue to operate with the support of its fan base who are rallying round the current share issue and a host of other fundraising events."
Scottish Cup holders Hearts, founded in 1874, have appealed to supporters to sign up to a share issue designed to raise £1.795 million and to buy tickets for home games starting with the match against St Mirren on Saturday.
At the weekend Hearts rejected a £500,000 offer from Rangers for early settlement of an £800,000 debt over player transfers.
Rangers are the biggest casualties of a financial crisis facing Scottish football. The Glasgow club are relaunching from the fourth tier this season after they collapsed under debts earlier in the year.
The Edinburgh club said in a statement on Tuesday night that they had been granted until December 3 to settle their debts, with their owed amount of £449,692,04 being split over two payments.
The club have also provided HMRC with assurances that future payments would be made in a "timeous manner".
Hearts were last Wednesday served with a winding-up petition and given eight days in which to pay what they owed, putting their future in severe danger.