Bury have been granted a 14-day extension after a winding-up petition brought by HM Revenue & Customs was adjourned in the High Court.
The club have been given the additional time to pay back smaller businesses, reports BBC Radio Manchester.
The Shakers were expelled from the English Football League in August as a result of their financial problems.
A prospective buyer for the club ended their interest on Monday, leaving Bury on the brink of liquidation.
A group of Bury supporters have already been working on plans to form a phoenix club and, if successful, would have to apply to the Football Association for entry into the English non-league pyramid next season.
Bury North MP James Frith met with the FA last week to discuss possible admission into the National League system, the fifth and sixth tiers of English football, and reported positive talks.
National League regulations state that its committee will determine “at its absolute discretion” in which league any new club shall be placed and will set out requirements to be met by the new club.
An initial application must be made by 1 March, with all necessary documentation submitted by 31 March.
Speaking to BBC Radio Manchester on Tuesday, Frith said: “In the event that the club is liquidated, we have to carry on with the plan which is an application to the National League system with a new incarnation of Bury Football Club.
“From my conversations with the FA, that can include what they call mitigation, which would mean our application would be considered alongside the size and heritage of our club.
“While liquidation would ordinarily mean entry to the lowest step in the pyramid, there is an outside chance with Bury’s circumstances – and this would be featured in any application we would make – being returned to an appropriate level within the National League system.”
From promotion to expulsion
Bury achieved promotion to League One last season amid a backdrop of severe financial difficulty, which had been ongoing even before Steve Dale bought the club for £1 from previous owner Stewart Day.
The club was up for sale again within five months, with Dale saying the problems “turned out to be far in excess of what we could have comprehended”.
A company voluntary arrangement (CVA) was agreed with creditors to help clear some of the club’s debts, resulting in a 12-point deduction.
Bury’s first six fixtures of the season were suspended while the EFL awaited evidence of how it would pay creditors and that it had the funding to complete the 2019-20 campaign.
When a last-ditch takeover bid by C&N Sporting Risk collapsed before the EFL’s deadline on 27 August, the league said it decided “after a long and detailed discussion” to remove Bury’s EFL membership “with enormous regret”.
It was the first club to have its league membership withdrawn since Maidstone were liquidated in 1992.
A “rescue board” – including local politicians and supporters’ group Forever Bury – made a proposal for the Shakers to be placed in League Two next season, but it “did not have the necessary support” and was rejected at a meeting of the EFL’s member clubs on 26 September.