And the administrator in charge of Ukio Bankas has warned the club would be liquidated unless the offers are improved.
The Lithuanian bank is owed £15million by Hearts but its administrator Gintaras Adomonis has issued the grave threat to BDO, the trouble-shooters now running the Edinburgh club after they were also placed into administration.
Adomonis has now told BDO to continue negotiation with fans' group the Foundation of Hearts and Five Stars Football Ltd, which lists controversial former Livingston owner Angelo Massone among its directors.
A third offer placed by HMFC Ltd has been rejected outright.
If the negotiations fail, Ukio Bankas will initiate the process of liquidating Hearts.
In a statement, Adomonis said: "I can repeat that we are doing everything we can to save the club functioning. However, I am obliged to protect solely the interests of Ukio Bankas and its creditors.
"If no feasible offer with terms and conditions acceptable to Ukio Bankas creditors is achieved, Ukio Bankas will remain with the only solution - liquidation of Hearts of Midlothian Plc and enforcement of the standard security over Tynecastle stadium.
"I sincerely hope this is the way of things we still can avoid."
Ukio Bankas holds a 29.9 per cent shareholding in Hearts and has a floating charge on Tynecastle as security against the money it is owed.
Meanwhile, Kaunas-based investment firm UBIG is owed a further £10million by the club. It owns a 50 per cent stake.
Both companies were once controlled by former Hearts owner Vladimir Romanov but are now insolvent. Ukio Bankas is in the hands of administrators while the process of appointing trouble-shooters to oversee the liquidation of UBIG was launched last week.
The statement released by Mr Adomonis added: "Hearts administrator BDO received three offers from investors, none of which were acceptable.
"Last week the potential investors presented improved offers, however, the main creditor's expectations were still not met. Ukio Bankas approved the administrator's proposal to continue negotiations with the two potential investors. The third investor's - HMFC Limited - offer was rejected."