If House Republicans get their way in budget talks, White House spokesman Jay Carney may be forced to make good on his pledge to sign up for Obamacare if his private insurance plan is dropped.

House GOP leaders have pressed during the government shutdown fight to significantly delay or defund the president's signature healthcare reform laws. But with those efforts failing to take hold, Republicans on Tuesday shifted fears and are seeking to extract a different health care concession: Forcing members of Congress, President Obama and top White House officials to enroll in the law's insurance exchanges without subsidies.

Carney is already on the record saying he would enroll if he loses his current employer-provided health care.

In late September, in anticipation of the Oct. 1 rollout of the law's online insurance exchanges, Fox News' Ed Henry asked Carney whether he would sign up for Obamacare.

“Absolutely,” said Carney. “If I did not have employer-provided health insurance ... then I would absolutely enroll.” He added that the Obamacare option would be more affordable than other plans in the private market.

House GOP leaders on Tuesday floated a broad proposal to reopen the federal government and raise the nation's debt limit. The plan also delays a tax on medical devices and would eliminate insurance subsidies for members of Congress. The plan was presented to rank-and-file members of the GOP caucus behind closed doors and leaders were revising the proposal ahead of a possible vote.

Regardless of the bill's final form, House Republicans insist they will make it a priority to ensure that Washington lives by the laws it passes. They say any bill they move must eliminate insurance subsidies for members of Congress, President Obama, Vice President Joe Biden and White House political appointees if they purchase health insurance under Obamacare's exchanges.

Majority Leader Eric Cantor, R-Va., said Tuesday that any budget solution must include what he called “our position on fairness” -- “no special treatment under the law.”

Congress has balked at having to participate in Obamacare after a provision from Sen. Chuck Grassley, R-Iowa, required 11,000 lawmakers and staff to lose the generous coverage they now have as part of the Federal Employees Health Benefits Program. Instead, they would get the lower-quality plan that offers fewer choices, known as “Medicaid Plus” in the exchanges.

Members and their better-paid aides also wouldn't qualify for Obamacare subsidies under the law, which means they could be exposed to thousands of dollars a year in out-of-pocket insurance costs.

After pressure from both parties, Obama ordered an exemption allowing congressional offices to continue to receive subsidies for insurance.

House Republicans are now pushing to put those subsidies on the cutting block in any deal to end the shutdown and avoid a default.