The Foundation of Hearts launched "one final push" for its bid at Tynecastle on Thursday morning, seven days before the deadline for first offers which was set by administrators.
About 4,000 fans have converted their pledges of support into direct debits, with several thousand more urged to follow suit before July 12.
The Foundation of Hearts says it has "two clear options" to fund a takeover. These will remain confidential until after the deadline but the group says both rely on securing as much financial support from individual fans as possible, coupled with partnerships with businesses.
They would also see a largely democratically elected Foundation of Hearts board control a majority stake, allowing it to install a football club board.
Foundation of Hearts chairman Ian Murray, the Labour MP for Edinburgh South, said: "We are nearly there. Should we be able to hit our own target for pledges we will in a position to draw down substantial capital to confirm a viable bid to take Hearts into the ownership of the people that matter most, the supporters.
"We now have two clear options negotiated and as a board, the Foundation of Hearts will select the most appropriate route, with a decision to be made in the coming days."
Former Hearts defender Dave McPherson has backed the deal.
"Now is the time," he said. "I want to offer my complete support to the Foundation of Hearts.
"I hope the Foundation can secure the very final pledges requires to make the sort of bid to the administrators that they consider most appropriate to the future of this club."