Administrator BDO confirmed it had concluded negotiations with its colleagues in Lithuania, who are dealing with the bankrupt financial institution.
The Foundation of Hearts had a Company Voluntary Arrangement offer accepted by the club's creditors on November 29 and is now a significant step closer to a takeover.
This will be completed by Bidco, a group of Hearts-supporting business people who have raised the capital for an immediate purchase, before the club is transferred gradually to Fanco, the 7,800 supporters who have begun paying monthly direct debit payments to the Foundation of Hearts.
A statement from BDO read: "The administrator of UBIG has confirmed they will transfer for an undisclosed sum their 78.97 per cent shareholding in Heart of Midlothian FC to Bidco to enable the CVA to conclude.
"This means that transfer of the club to Bidco can now go ahead with a timescale of eight to 10 weeks for completion of the sale and purchase agreement subject to the deal being ratified through the Lithuanian courts and/or creditors."
Joint administrator Bryan Jackson added: "This is welcome news for the club as it means that the transfer of ownership can go ahead. While there remain some technical issues to resolve the process is now gaining some momentum.
"I would again like to thank the fans, the staff and the players for their continued patience during this time. For all of those involved with Hearts, on and off the pitch and on the terraces, it has been a very difficult period and their assistance and support has been greatly appreciated."
The news is a much-needed lift for the Hearts fans after their team's League Cup semi-final shoot-out defeat by Inverness on Sunday, which came after they led nine men with 94 minutes gone.
Foundation of Hearts chairman Ian Murray said: "This is very welcome news from BDO. We are edging closer to getting the club out of administration and we are now talking about when rather than if.
"This is a very significant hurdle overcome and we are delighted but more work is required.
"Let me continue to stress that we encourage every supporter who has not yet done so, and is able, to set up their monthly contribution at www.foundationofhearts.org."