The joint call from the key supporters clubs to 100,000 fans came after a two-hour meeting with Tynecastle executives David Southern, Sergejus Fedotovas and Vitalijus Vasiliauskas. After the talks the directors said: "While there is a need for an immediate short-term cash injection, there is no quick fix."
Dunfermline confirmed they had just 12 days to raise £81,000 owed to HMRC as it was also announced that John Yorkston, chairman of the club for the past 14 years, will be stepping down from his post at the end of the current campaign.
A study by analysts Begbie Traynor earlier this week found one in five top clubs out of 32 in Scotland's top three divisions, could fold by the end of the season, but they did not reveal which clubs were most likely to go first.
As Hearts and Dunfermline address their money problems, Falkirk have also been reported as facing financial difficulty.
In the capital, a joint statement was released following crunch talks at Tynecastle attended by the Federation of Hearts Supporters Clubs, Heart of Midlothian Shareholders' Association, Hearts Supporters Trust and Hearts Youth Development Committee.
The groups' statement said: "It is clear to us that the club needs more than ever to raise immediate finance through the share offer and buying match tickets for at least the next three home matches to secure our short-term future and, therefore, our long-term sustainability.
"By all supporters, we mean all those who attended Hampden on May 19 [for the Cup Final] and those who lined the streets of Edinburgh to enjoy the cup parade. This is a call to arms."
Steve Kilgour, of the Federation of Hearts Supporters' Clubs, urged fans to push for their own 10% stake in the club by raising £2m in just five months, bidding to raise £450,000 for a tax bill by next week.
Mr Kilgour said: "Vladimir Romanov is no longer financing the club or bankrolling it. We need £2m to carry us through to the summer."
One fans' website, Jambos Kickback, has already bought 50,000 shares in the club worth £5500 after supporters donated cash.
A club spokesman said: "Our battle has well and truly begun in order to ensure that we can reach the end of the season.
"If we can achieve this goal then we have a very good chance of ensuring Heart of Midlothian Football Club can avoid repeating events as the club will have a real basis on which to operate self-sustainably."
The Rangers crisis coupled with poor attendances and a reduction in television revenue are said to have had an impact on many of Scotland's top-flight clubs, but the news Hearts had been served with a winding-up order has shocked many supporters.
Just four months ago, Edinburgh had welcomed home from Hampden Park in Glasgow the newly crowned Scottish Cup winners.
Residents are rallying round the club, with Edinburgh bar The Brauhaus hosting a fundraising event tomorrow, with all the takings going towards buying shares.
Manager Salman Sarwar said: "We hope to raise around £2000 at the bar and we will also auction off various famous shirts, signed memorabilia and other things, which will all go to the club.
"If any Hearts fans, or Scottish football fans in general, are out for a pint on Sunday I would implore them to go to The Brauhaus, as every drop they drink will help Hearts Football Club."
Edinburgh Capitals ice hockey club are offering Hearts season-ticket holders 25% discount on their next home fixture, with all proceeds being donated to club.
Fundraising efforts were described by the club as a magnificent gesture.