“What Republicans have to realize [is] the political winds are in our direction. We can’t risk changing the winds at this stage,” said GOP strategist Matt Mackowiak. “You can shut down the Obama agenda completely if you have the Senate.”

A GOP decision on the debt ceiling is likely to come out of the annual House retreat scheduled for Jan. 29 in Maryland. Treasury Secretary Jack Lew has told Congress he expects the government could be in danger of missing payments by early March, giving lawmakers a narrow window to find another borrowing boost.

“The only way you lose the House is if Dems intercept a Hail Mary pass on the debt ceiling,” another GOP strategist said. “If you lose the House, you open the door to everything.”

At the same time, GOP tacticians acknowledge that the party cannot look weak in the debt ceiling fight and simply grant the president the “clean” hike he wants. Furthermore, the timing of the fight means members facing conservative primary challenges will face a tough dilemma.