Fans' group the Foundation of Hearts has been named preferred bidder for the club.
Hearts entered administration in June and BDO, which has taken over the running of the club, revealed last month that
club debts amount to £28.4m.
Two other parties submitted formal offers for the club.
"This is an important day in the pathway towards putting Heart of Midlothian back on its feet," said the foundation's chairman, Ian Murray MP.
Hearts boss Gary Locke
“It'll be Hearts people running the football club and that's the most important thing”
"We are proud that the bid backed by the Foundation of Hearts has been named preferred bidder.
"Let us be clear though. The journey is only beginning. Our bid has not yet been accepted, albeit we are hopeful an agreement can be reached.
"Now, for the first time, the fate of Heart of Midlothian really is in the hands of us, the supporters of this remarkable football club."
Hearts manager Gary Locke welcomed the news and urged fans to support the foundation's bid.
"It's great news, fantastic news," he said. "The fans have been unbelievable and we're asking them to put their hands in their pockets once more.
"But, if they can come on board, it will leave the club in a stronger position.
"Obviously we've got a long way to go, but now we know the Foundation of Hearts are the preferred bidder - they're Hearts people and have the club at heart - that's the news myself and all the playing staff were looking forward to hearing.
"It's a big day for the football club and hopefully now we can move forward.
"It'll be Hearts people running the football club and that's the most important thing."
Any deal to exit administration via a company voluntary arrangement (CVA) must be agreed with the major creditors of the club.
“This is a positive development but does not guarantee that a CVA will be successful”
Objections have been lodged in a Lithuanian court over the proposed liquidation of Hearts' parent company, UBIG.
The Lithuanian investment company owns 50% of Hearts shares and, until the court makes a decision on the future of the company, no sale of Hearts can be completed.
A decision from the courts on UBIG could be made before the end of September.
Lithuanian bank Ukio Bankas, itself in administration, is Hearts' other main creditor.
Bryan Jackson, of BDO, commented: "We all still have a lot of work to do to demonstrate to the major creditors that the bid can offer them the best possible outcome. Let's not get carried away - now is the time for everyone to get behind FOH to make this bid work.
"This is a positive development but does not guarantee that a CVA will be successful. That will require a considerable combined effort from FOH and BDO to ensure that all interested parties are satisfied.
"At the moment, we cannot put a timescale on the process as a number of negotiations with various parties still need to take place and we are waiting for the appointment of administrators to UBIG to be ratified.
"However, we can ensure that collectively we make as much progress as possible on our side so we are able to finalise the matter quickly once the opportunity presents itself."
The Edinburgh side started the season with a 15-point deduction due to their insolvency problems, but the weekend win over city rivals Hibernian has reduced their points total to -12 after two Scottish Premiership fixtures.