The club entered administration in June, incurring a 15-point penalty which has contributed to their relegation from the Scottish Premiership.
On Wednesday, creditors of Ukio Bankas agreed the sale of the shares, with businesswoman Ann Budge ready to take control of the club.
Budge, of holding company BIDCO, will gradually hand over control at Tynecastle to fans group Foundation of Hearts.
Ian Murray, MP for Edinburgh South and a member of the group, confirmed the positive news as the club look set to secure the first step in their financial recovery.
"This is absolutely fantastic news and a great development," he said.
"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."
