Budge says Hearts are 90 per cent towards their goal following a breakthrough on Wednesday when the creditors of Ukio Bankas, the club's main creditor, approved the £2.5million deal and transfer of shares.
Championship-bound Hearts had already secured approval from major shareholder UBIG, but there is a cooling-off period until April 27 during which one of its creditors could object.
The sale and purchase agreement between Budge's Bidco 1874 company and UBIG cannot happen until then but Budge cautioned that other documentation needed to be finalised in Scotland, including the agreement that will see Budge ultimately hand over control to the Foundation of Hearts supporters group.
Budge said in a letter to fans: "Since we are all on the same side in this, then all of this work is being done in the spirit of co-operation.
"However, all parties interests must be protected - in particular, the fans who are putting up the cash to help stabilise the club and thereafter, to buy the Hearts shares and Tynecastle Stadium, taking it out of private ownership and into fans ownership, hopefully for all time.
"This is a major undertaking and it is crucial that the legal work is completed to the highest standard. We cannot cut any corners in this, so please bear with us as we work to get this all done."
Budge also pointed out that she was obliged under stock market rules to make an offer for all other shares under the same terms as the deal offered to UBIG.
"The whole process will be fully explained at the time," she said.
"However, for legal reasons, this will delay the formal leaving of administration for a few more weeks."
Budge added: "We are all very confident that this work will be completed well before the start of next season to avoid the possibility of any further football sanctions. I am also working with the current management team at Tynecastle, to ensure I have a clear understanding of the issues, to allow me to 'hit the ground running' in terms of taking over control of the club, when legally allowed to do so."