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But The Herald has learned that a deal unrelated to the club must be completed to generate the £5million-plus needed for the takeover.
Mr Heaney's lawyers have written to administrators saying completion of that is "imminent".
But the figures at the centre of the takeover have already taken steps towards extending their deadline, in case that should become necessary.
The Herald understands Mr Heaney is working on a lucrative deal linked to an expanding development firm.
Lead administrator Brendan Guilfoyle admitted: "We're told there is a deal he's completing at the moment that requires completion to complete our deal."
But he stressed: "Nothing he says causes me any concern.
"We continue to look him in the eyes and we continue to ask him whether he'll complete.
"He is quite calm about it so we are just pushing forward. All I'm told is there is an enabling deal still to complete; he's got something coming to the table which enables him to move on to this."
Mr Heaney's solicitors have confirmed in writing that alternative funding avenues remain open, Mr Ridsdale added.
"He's got a deal going on that is nothing to do with Argyle and we've been given assurances it should be completed in a week or so," he said. "That's how he'd prefer to fund it but, if not, he has other routes to which he can fund to the level required.
"In all the conversations we've had, we've had no indication that the preferred funding route is the only funding route.
"I'm relaxed about it. You don't get to this stage, put in all this work and then not complete."
The council has strict guidelines on what is allowed to be developed in Central Park.
The Herald revealed in March how Mr Heaney was in talks with cinema operator Cineworld, which will not comment.
He was previously considering student accommodation, since ruled out, but is now exploring potential catering opportunities.
Mr Heaney has been unavailable for comment, but both Mr Ridsdale and Mr Guilfoyle are confident the takeover will be completed by the end of the month.
A sale and purchase agreement giving Bishop International exclusivity with administrators expires on August 12 – but contains a clause allowing all parties to extend the deadline if completion is close.
As part of the deal the company, whose owners are concealed behind nominees, must complete a deal over secured £2.1million debts with mortgage lender Lombard.
They will then give Mr Ridsdale enough cash to settle Argyle's £3.2million football creditor debts and cover this season's predicted shortfall of around £1million.
Football League regulations ban financial interests in more than one club – and the governing body has the power to block any takeover that breaches them.
But Mr Ridsdale added: "It will be up to me, as chairman, how I manage the club with that money.
"[Bishop International Limited] won't have financial influence because they can't dictate how that money is used."
Meanwhile, Devon entrepreneur James Brent was due for a third 'Rescue Plan' meeting with supporters' groups today.
The package has been launched amid fears failure to complete the current bid could drive the Pilgrims out of business.
Mr Guilfoyle has already provided financial information to the group in case help is needed.
But he said: "We told the fans group that – if it's a rescue bid and if the current preferred bidder doesn't complete – they have to be ready with funding."