The Ohio Republican and other House leaders have repeatedly noted that the chamber passed two sequester substitutes last year, and they have used that fact to challenge the Senate to act. But the House has not yet approved any legislation in the new Congress to avert the more than $1 trillion in across-the-board spending cuts that will begin Friday.

The previous measures passed during the 112th Congress and would have to be passed again before the Senate could even consider taking them up. But even in the 112th Congress, both bills squeaked by with few votes to spare. That has led some sources to speculate that if the House brought up a similar bill again, it would quite simply fail.

“All the momentum in House Republican circles now is for keeping the sequester as is. I don’t see how any sequester replacement bill could get any oxygen for the foreseeable future,” one Republican aide said.