Trafford Council is the other, and two years into receiving EFS, residents have seen council tax increases of 7.49% each year.
Finance leaders at Bury have stated they do not believe borrowing was the solution for the council, but did not rule out tax rises of more than 4.99% if the government allowed it.
“A lot of the decisions we’re making, and even the ones that we haven’t made yet, are all regrettable,” said Sean Thorpe, the council’s lead member for finance.
“Clearly we’re at the stage now as a council where we’re thinking the unthinkable and know that won’t be enough.
“I think we’re in a position where we’ve got to look at all options, no matter how distasteful they are.”
He added the council was aware people were already struggling but, because services were declining, a higher council tax was something the authority would have to consider.
The council said EFS would be needed for three years while it took steps to bring down its spending, saying projections did not show it “going viable any time soon”.
Among the plans floated by the authority were the introduction of green waste charges and cuts to youth services, as well as switching off streetlights between 12am and 5am.
The council is also currently engaged in a voluntary redundancy scheme for staff.