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House Republicans are teeing up a vote this week on a new debt ceiling bill, marking the first legislative battle of President Obama's second term and one that could determine whether the country once again risks default over a political fight.

House leaders, after unveiling the legislation Monday, are planning to hold a vote Wednesday on their plan to allow the government to keep borrowing through May 18.

While the short-term increase is getting mixed reviews, the second plank of the legislation -- meant to pressure Senate Democrats to pass a budget -- has also raised questions.

Under the proposal, Congress would withhold the pay of lawmakers in either the House or the Senate if their chamber fails to pass a budget by April 15. House Republicans have passed budgets for two consecutive years, but the Senate hasn't passed one since Obama's first year in office.

But the so-called "no budget, no pay" provision has run into complaints that it's not constitutional. Critics point to the 27th Amendment, which states: "No law, varying the compensation for the services of the Senators and Representatives, shall take effect, until an election of Representatives shall have intervened."

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Rep. Hakeem Jeffries, D-N.Y., said Monday "it appears that the 27th Amendment does not permit Congress to alter its pay in the midst of a current session."

House Republican leadership is defending the plan. A GOP leadership aide told Fox News that members would ultimately get paid, since "constitutionally we have to pay members." The suggestion is that the plan would not violate the 27th Amendment because it would temporarily withhold pay, as opposed to reducing pay. Under the plan, members' pay would be put in escrow starting on April 15 for any chamber that hasn't passed a budget resolution.

As for the debt ceiling provision, the legislation does not set a specific limit; rather it would automatically increase the limit by the amount required to fund U.S. government obligations through May 18.

White House Press Secretary Jay Carney said Tuesday that Obama would not block the bill if it passes.

Democrats in Congress have generally reacted coolly to the three-month extension, which would still leave other choke points in place, including sharp, across-the-board spending cuts that would start to strike the Pentagon and domestic programs alike on March 1 and the possibility of a partial government shutdown with the expiration of a temporary budget measure on March 27.

But failing to meet those deadlines would have far less serious consequences than defaulting on U.S. obligations like payments to bondholders.

Obama has vowed he would not bargain over the debt limit. If the debt cap is not raised, the government would default on its obligations by as early as Feb. 15, Treasury says.

House Republican Leader Eric Cantor confirmed to Fox News that the House will vote on the package Wednesday.

The debt-ceiling bill stands as the first legislative test of Obama's second term -- the president took the public oath of office Monday during the inaugural ceremony in Washington.

On the heels of the hard-fought deal to avert the fiscal crisis, the debt ceiling, the automatic spending cuts and the expiration of the current budget are the immediate domestic challenges facing the administration and Congress. All three are also opportunities to tackle the country's bloated deficit -- though the president in his inaugural address pushed back on calls to cut entitlements.

"We must make the hard choices to reduce the cost of health care and the size of our deficit," Obama said, speaking to the hundreds of thousands of people watching his speech on the National Mall. "But we reject the belief that America must choose between caring for the generation that built this country and investing in the generation that will build its future."

The president has a few more inaugural obligations to complete before getting down to business. He and Vice President Biden were attending a prayer service Tuesday morning at the National Cathedral, and they will celebrate with campaign and White House staffers at another glitzy inaugural ball Tuesday night.

Otherwise, Tuesday is expected to be a normal working day at the White House. The president will meet with top aides, and press secretary Jay Carney will brief the press.

Behind the scenes, Obama and his advisers are working on plans to unveil a comprehensive overhaul of U.S. immigration laws, which is expected to be a central topic in Obama's Feb. 12 State of the Union address. The president also will be seeking support from a wary Congress for the far-reaching package of gun control proposals he unveiled last week, including an assault weapons ban and universal background checks for gun purchasers.

The Associated Press contributed to this report.