That means the Senate would remove any provisions to defund Obamacare and send the stripped-down spending proposal back to the House.
Boehner would then have to decide whether to put it to a vote, even though that could undermine his already weakened leadership by having the measure pass with only a few dozen moderate Republicans joining Democrats in support.
If he refuses to bring the Senate version to the floor for a vote, a shutdown would ensue.
"The key player is really Boehner," West said.
Polls showing a decrease in public support for the health care reforms embolden the Republican stance. Meanwhile, surveys showing most people oppose a government shutdown and that more would blame Republicans if it happens bolster Democratic resolve.
Voices across the political spectrum warn against a shutdown, including Congressional Budget Office Director Douglas Elmendorf, Federal Reserve Chairman Ben Bernanke, the U.S. Chamber of Commerce and Republican strategist Karl Rove.
"Even the defund strategy's authors say they don't want a government shutdown. But their approach means we'll get one," Rove argued in an op-ed published Thursday by the Wall Street Journal.
He noted the Democratic-controlled Senate won't support any House measure that eliminates funding for Obamacare, and the White House said Thursday that Obama would veto such a spending resolution.
"Republicans would need 54 House Democrats and 21 Senate Democrats to vote to override the president's veto," Rove noted, adding that "no sentient being believes that will happen."
West concurred, telling CNN that "you can't expect a president to offer his first born to solve a political problem for the other party."
"It's the House split that's causing this to happen," he noted. "People now equate compromise with surrender. It's hard to do anything under those circumstances."
Under the compromise scenario, the Senate would remove provisions defunding Obamacare from what the House passes while perhaps making other relatively minor changes to provide Boehner and House Republicans with political cover to back it.