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Mr. Barrasso and Mr. Hatch were among the sponsors of recently filed legislation to repeal the health insurance tax, an annual fee that taxes health insurance providers relative to the worth of the insurance premiums they collect each year. It affects providers who bring in more than $50 million the hardest and will phase into effect from 2014 to 2018, according to the Americans for Tax Reform.

Mr. Barrasso also took to the Senate floor in February for a told-you-so speech about unions, citing a Wall Street Journal article that said some labor groups have expressed buyer’s remorse over parts of the health care law they supported.

“Since unions supported the president’s health care law, it’s ironic that unions are now speaking out against parts of it and asking for special loopholes,” the senator’s spokeswoman, Emily Lawrimore, said. “Now that they’ve ‘found out what’s in it,’ like most Americans, they don’t seem to like it.”

The latter comment is a dig at House Minority Leader Nancy Pelosi, California Democrat, who suggested in March 2010 that Republicans needed to pass the massive health care bill to see what it contains. What Americans are seeing is not pretty, Republicans say, but their hopes to claw back the law remain an uphill battle.

“As I see it,” Mr. Aaron said, “the chances that any law could pass that delays implementation of the [health care] law as it stands are currently nil.”