Camp said he does not support a debt-limit hike that doesn't also require "significant spending cuts and budgetary reforms."

"Increasing the debt limit without showing that we can achieve real spending restraint would likely lead to very similar results as default, a lower credit rating, higher borrowing costs and more expensive imports," he said. "Such irresponsibility would most certainly increase the cost of oil and gas, making the pain at the pump that much worse."

While the anticipated House rejection of the bill is meant to send a political signal that President Obama's budget proposal cannot be passed, Republicans are expected to increase the debt limit later this year, even if the increase is less than $2.4 trillion.

Ways & Means Ranking Member Sandy Levin (D-Mich.) said the GOP bill is a "dangerous political stunt," and said members should instead focus on bills that can pass.

"We should set a framework for immediate and long-term deficit reduction, but not by ending Medicare and doubling seniors’ health care costs, as Republicans have advocated," Levin said. "And not by threatening a default that numerous economists and CEOs have warned would have calamitous consequences and jeopardize our economic recovery."

-- This story was updated at 2:05 p.m.