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Mr. Lee said roughly a dozen colleagues have signed on to a letter addressed to Mr. Reid that lays out his proposal to reject any appropriations bills or continuing resolutions that contain funds for the health care law.

“If the administration will not enforce the law as written, then the American people should not be forced to fund it,” the letter says.

The Club for Growth, a conservative political action committee, called on Mr. McConnell on Wednesday to sign the letter and “encourage others to sign on.”

A spokesman for Mr. McConnell referred questions about the minority leader’s comments Tuesday in which he said only that the party is discussing options internally and is focused on cutting $2 trillion over 10 years in line with a 2011 deal to reduce spending.

It’s unclear whether conservatives will garner enough support for Mr. Lee’s measures to mount a filibuster threat in the Democrat-controlled Senate, but the senator from Utah pointed to three rising stars in the Republican Party — Mr. Rubio, Ted Cruz of Texas, and Rand Paul of Kentucky — as key allies in his effort.

But the concept could be a tough sell within the party.

Sen. Lindsey Graham, South Carolina Republican, said he fully supports defunding and replacing the health care law, but would not tie it to the spending resolutions that fund all government operations because it would put other priorities at stake.

“We’ve got a military under stress, and you’re talking about a construct where they wouldn’t get paid,” he said. “The Iranians would support that idea. I don’t.”

Mr. Rubio said Democrats would take the heat if the dispute brings Washington’s fiscal house to the brink.

“The way I see it is President Obama and his allies are willing to shut down they federal government unless Obamacare is funded,” he said. “I just don’t think Obamacare is more important than America.”