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A second Democratic aide said how the party handles the issue — such as whether Democrats will propose a motion to recommit — will have to wait until it knows what Republicans bring to the floor.
"Our approach is going to depend on how they solve their internal problems," the aide said.
Democrats also said there is less pressure on them, given that the amendment would need scores of Democrats to pass.
Ryan called it "another parlor trick by the Republicans," and he said Democrats in tight races do not fear the vote because "their policies and lack of regulation of Wall Street collapsed the economy, and we're still cleaning up their mess and they're pointing fingers at us. So the facts don't bear out their accusations."
In the Senate, Minority Leader Mitch McConnell (R-Ky.) will force a vote on the amendment next week.
"The time has come for a balanced budget amendment that forces Washington to balance its books," McConnell said on the floor Wednesday. "If these debt negotiations have convinced us of anything, it's that we can't leave that to the politicians in Washington to make the difficult decisions they need to get our fiscal house in order."
A GOP aide said that although it is unlikely to get the 67 votes needed to pass, it will put Senators on record. "If you are serious about reducing spending you would support it," the aide said.
Jessica Brady contributed to this report.