Administrators BDO said the Foundation of Hearts have put together a 'credible' attempt to save the SPFL side from debts totally almost £29m.
The development looks likely to signal the end of the bid from Five Stars Football Ltd, who list former Livingston director Angelo Massone among their directors.
Bryan Jackson of BDO said: "The naming of a preferred bidder is the next step in the process of selling the club.
"It allows us to work with one party around which everyone can rally. In doing so, we maximise the prospects of agreeing a sale price that is acceptable to the Administrator of Ukio Bankas, and securing the necessary consent and ownership for FOH.
"This is an important milestone and reflects our belief that FOH is a credible bidder for the club. However, 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 administrators of Ukio Banka - the Lithuanian firm which are owed £15.5m but still hold a 29.2 per cent stake in the club, told both bidders last month that their offers were 'well below the level of finance needed to dissuade them from liquidating Hearts and using the money to settle some of the insolvent investment firm's £380m debts.
After requesting further offers from the two remaining interested parties and discussing them with the administrator of Ukio Bankas, BDO has decided FoH's offer is the best hope for the club.
As well as debts to Ukio Bankas, another firm once controlled by former club owner Vladimir Romanov - Hearts majority shareholder UBIG - is due £8.2m, with around £5m due to a list of other companies.
Ian Murray MP, chairman of Foundation of Hearts, said: "This is an important day in the pathway towards putting Heart of Midlothian back on its feet.
"We are proud that the bid backed by the Foundation of Hearts has been named preferred bidder."
Hearts boss Gary Locke hopes the latest developments is the first major step in the club's road to recovery.
"It's fantastic news," he said. "I know it's a big ask again because the fans have been unbelievable and it's asking them to put their hands in their pockets once more but if they can all come on board, it will certainly leave the club in a stronger position.
"We've still got a long way to go but now we know the Foundation is preferred bidder. They are Hearts people and they've got the club at heart. It's the news that myself and all the playing staff and the staff behind the scenes were looking forward to hearing.
"It's a big day for the football club and hopefully now we can move forward."
Locke says he is happy with the prospect of a fan ownership model being in place at the club.
He added: "It has happened in other countries, with Germany probably being the most notable. Portsmouth as well. It seems to work. For me, it's great in as much as it's going to be Hearts people who are going to be running the football club and I think that's the most important thing."