The Lithuanian bank is the largest creditor of Hearts and the company which holds security over Tynecastle Stadium and a 29.5 per cent shareholding, which will now be sold for £2.5million to businesswoman Ann Budge following agreement at a meeting of Ukio Bankas' creditors committee.
A source close to the Gorgie club said: "Like any of these processes there may still be a few creases to be ironed out, but this is the club effectively safe from liquidation.
"If everything goes to plan, the club should formally exit administration sometime next month."
Vaidotas Vysniauskas, spokesman for Ukio Bankas's administrators, said: "The creditors committee of Ukio bank have approved it today and it is a final step from Ukio bank, fortunately."
Millionaire former IT specialist Budge, 66, has fronted the cash for Bidco, the company which will now take over the running of the club.
Budge will become executive chair of the club "on a no-fee basis" before transferring ownership of the Jambos to fans group the Foundation of Hearts within five years of gaining control.
In the meantime, the Foundation's 8,000-plus membership has pledged to donate cash on a monthly basis that will be used to pay for the club's running costs before Budge is given back the £2.5million.
Hearts went into administration in June last year and, as recently as April 1, FoH chairman Ian Murray MP admitted the club was in "serious danger" of being liquidated as the creditors process with UKIO and UBIG - the club's other major shareholder - in Lithuania dragged on.
But he said: "UKIO Bankas creditors met this morning and approved the deal. A Sale and Purchase agreement needs to be completed but we can say 'deal done'.
"This is absolutely fantastic news and a great development. It is vindication of our wonderful supporters' decision to support the club during the most difficult times. Without them Hearts would not be here.
"We are 90 per cent there, pending the successful completion of a sale and purchase agreement.
"I want to once again thank the fans but emphasise that financially this will go right down to the wire. I encourage every Hearts supporter to fill every seat at Tynecastle between now and the end of the season."
Administrator Bryan Jackson of BDO added: "I am delighted that the creditors' committee has today agreed the transfer of shares to Bidco. This is one of the final hurdles in the transfer of ownership of the club.
"I would like to thank the creditors of both UBIG and Ukio Bankas for agreeing to this deal and also the patience and forbearing of Ann Budge and the Foundation of Hearts.
"I appreciate that many fans will have wondered whether this day would ever come but we have been working tirelessly on behalf of the club to make this deal work."
"There is still some work to be done to conclude proceedings but we are now very close to a successful conclusion of the CVA.
"However, I would caution that there is a cooling off period for the consent for the UBIG creditors which ends on April 28, although not one for Ukio as they have a creditors committee, and we are working to have the sale and purchase agreement concluded, hopefully, before the end of the season."
Gintaras Adomonis, bankruptcy administrator of Ukio bank, also said: "The deal is very significant for us in several aspects. For the fans of Hearts this has been a very emotional and difficult period and we are happy that this has now been resolved in an amicable way."