BDO administrator Bryan Jackson confirmed that creditors of the club's major shareholder, UBIG, had agreed, by majority, to back the agreement with the Foundation of Hearts at a meeting on Monday morning.
BDO will now attempt to persuade the creditors of Ukio Bankas to approve the deal to take the club out of administration.
Failure could see the club move towards liquidation - two days after they were relegated from the Scottish Premiership.
However, it is unclear when the Ukio creditors will meet. There are some suggestions from Lithuania that they will not convene for another two weeks despite expectations the process would be completed on the same day as UBIG.
Foundation of Hearts chairman Ian Murray previously warned that a major delay could force the club towards liquidation.
The positive news from UBIG came after reports emerged of a rival 11th-hour bid from Pat Munro, who has failed in several previous attempts to buy the club in recent years.
However, the offer looks set to be dismissed without consideration.
A spokesman for the Ukio Bankas bankruptcy administrator, Gintaras Adomonis, confirmed that he had received an email from Munro, who reportedly offered £1million a year for 15 years.
But he added: "The form and content of it is beyond consideration - all financial proposals should be provided to BDO, the administrator of Heart of Midlothian."
Ukio holds a security over Tynecastle for the debt it is owed and nearly 30 per cent of shares, while UBIG owns almost 50 per cent of shares and is also a significant creditor.
A £2.5million Company Voluntary Arrangement offer was accepted in November last year while the Foundation of Hearts secured a deal to acquire UBIG's share earlier this year.
However, because both Lithuanian financial institutions are in administration, like Hearts, following the collapse of Vladimir Romanov's business empire, the takeover deal needs approval from their creditors as well.
Speaking last week ahead of the meeting, Murray said: "We just have to see what happens on the seventh and if it's delayed a week then that's something we can probably cope with.
''If it's delayed six months, then it's probably something that gives us greater problems."
The Foundation of Hearts, which has the financial backing of about 8,000 supporters, plans a gradual takeover by fans after Edinburgh businesswoman Anne Budge initially puts up the £2.5million needed to rescue the club.
Hearts went into administration in June last year with debts of about £30million. They were deducted 15 points and hit with a transfer embargo and went down on Saturday despite a 4-2 win over Partick Thistle.
Despite claims from those working on the Ukio administration that creditors would not give their decision for about two weeks, the Foundation of Hearts tweeted: "We understand the UKIO meeting is being held today and has started."