Bolton Wanderers have been given 14 days by the EFL to meet all outstanding requirements of the League’s insolvency policy or its membership of the Football League will be withdrawn.
The League One club had been given until 17:00 BST on Tuesday to complete the deal, having been granted an extension to Friday's initial deadline.
Bury are the first team to drop out of the EFL since Maidstone's liquidation in 1992.
League One will comprise of 23 clubs for the rest of the season, with only three teams to be relegated.
"Today is undoubtedly one of the darkest days in the league's recent history," said EFL executive chair Debbie Jevans.
"I understand this will be a deeply upsetting and devastating time for Bury's players, staff, supporters and the wider community. There is no doubt today's news will be felt across the entire football family."
Bolton Wanderers, also of League One, have been given 14 days to avoid being expelled themselves, with their prospective takeover by Football Ventures (Whites) Limited still yet to go through.
In a statement on Monday the club's administrators warned they are on the brink of liquidation, saying if they were unable to resurrect the move the club was "not in a position to carry on trading".