Arlene Foster was speaking after a meeting of the executive on Thursday afternoon.
She told the BBC there were no solutions on the table.
The Stormont parties failed to pass the Welfare Reform Bill on Tuesday.
Mrs Foster said failure to pass the legislation meant that there was a £600m hole in her budget.
She said she cannot move the budget bill required to give Stormont officials the legal authority to carry on spending money after the end of July.
The minister said she is seeking a meeting with the Treasury in London early next week to assess her options.
Sinn Féin's John O'Dowd said Stormont ministers should present a united front and seek a joint meeting with Prime Minister David Cameron.
However, both the Alliance Party and Ulster Unionist ministers said they were pessimistic about the chances of a resolution to the executive's difficulties.
The finance minister has said she cannot move the budget bill forward because of the inability of the Stormont parties to resolve the deadlock over welfare reform.