Ann Budge's BIDCO 1874 - the company fronting the bid to secure a controlling stake in the club - purchased UBIG's 50% following a creditors meeting on Monday, subject to a 20-day cooling off period.
However, Ukio's 29.9% shareholding and security over Tynecastle has not yet been attained, and Jackson intends to stay in Lithuania until Ukio have accepted BIDCO's £2.5m offer.
It is understood Hunt has pledged to do anything he can to help, should political intervention be required, with a source describing the talks as "extremely positive".
While reluctant to confirm the date, sources in Lithuania expect the pivotal meeting with Ukio to decide the club's future to go ahead within seven days and BDO are already working towards a draft timetable.
However, even if the share transfer from Ukio to BIDCO is ratified, it is not likely to be completed until mid-May due to the legally required 20-day cooling off period - beyond the end of April cut-off point BDO have estimated they will run out of money.
To keep Hearts solvent until the takeover is completed, the club's administrators are considering asking the Scottish Professional Football League for an advance of their end-of-season prize money.