[Washington / Fox News] -- 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."

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."