They are one of 11 clubs who have applied, but must meet certain criteria by 31 March to be considered for a 10th-tier place.
According to the BBC, it is understood a ground to play at next season has been found which meets these requirements.
The NWCFL lists them as one of two new clubs applying to play in the league next season, the other being FC Isle of Man, with the other nine teams coming from lower down the non-league pyramid.
“This confirms our belief that we are the only team currently with an application in to play football in Bury next season,” Bury AFC said on Twitter.
The league confirmed in December that an application had been made by the new club, which is unaffiliated to Bury FC.
At the time, prospective investor Robert Benwell said it would cost over £500,000 to start a new club in the lower divisions of English football.
Last week Bury FC, which still exists as a company, defaulted on a plan to help settle longstanding debts with its creditors.
The Shakers were due to play in League One this season but their opening fixtures were postponed and their expulsion from the English Football League came after a last-ditch takeover bid collapsed.