Hearts have a Friday deadline to pay £450,000 tax bill or face a winding-up order from Her Majesty's Revenue and Customs.
Reports suggest a consortium of Edinburgh businessmen are preparing to make a move to buy the club from majority shareholder Vladimir Romanov.
A £1.75million tax case was also revealed when a £1.79m share issue was recently launched, a scheme which could result in fan ownership.
"Hearts may be the third biggest club in Scotland so it would be a disaster," Burley told Sky Sports News.
"Hopefully they'll come through it and build from there. I spoke to one or two people at Hearts, in the boardroom, on Saturday. I think they're optimistic.
"A club like Hearts, such a big club, I'm optimistic things will come right."
Former Hearts player Gary Mackay is one of many aiming to raise funds and has claimed a contribution from Harry Redknapp, who took Tottenham to Tynecastle in the Europa League in August 2011.
Mackay told Sky Sports News: "Scott Crabbe and I, ex-players, have set up the 1874 fighting fund and the response we've had has been wonderful.
"A lot of ex-players and Harry Redknapp, the ex-Tottenham Hotspur manager, they've all pledged #100 to kick start that fighting fund.
"We'll pick up debts, because there are one or two debts and we'll have money in our coffers to work with on a weekly, monthly, yearly basis."