House Republicans' sweeping legislative replacement for the 2010 Dodd-Frank financial reform law cleared the last hurdle en route to the House floor Tuesday evening, as the House Rules Committee cleared it for debate.

The House is expected to vote Thursday on the measure, the Financial CHOICE Act, which would eliminate or roll back many of the post-crisis banking rules, and provide extra relief for banks that choose to maintain much higher levels of capital. The bill, H.R. 10, would also limit the discretion of regulators, dramatically reduce the powers of the Consumer Financial Protection Bureau, and implement reforms to the Federal Reserve's regulatory and monetary policy powers.

In a statement late Tuesday, the White House, previously quiet on the bill, said that it "supports passage" of the legislation "as a necessary and important step in moving financial reform legislation through the Congress. Upon passage, the administration looks forward to working with the Senate on arriving at a final piece of legislation."

Testifying Tuesday at the Rules Committee hearing, bill author and House Financial Services Committee Chairman Jeb Hensarling of Texas argued that his legislation would boost economic growth by increasing lending. "We have way too much capital sitting on the sidelines of this economy," he said.

Republicans have acknowledged that the conservative overhaul cannot pass the Senate, where Republicans must contend with a Democratic filibuster. Nevertheless, Hensarling explained Tuesday morning in a CNBC interview, passing the legislation creates momentum for a GOP conference that so far has yet to put major legislation on the president's desk. "I stand ready to negotiate in good faith with Democratic colleagues in the Senate," Hensarling added.

Democrats, meanwhile, have mounted a campaign of criticism against the bill, which they say risks another crisis by cutting back rules and is meant to benefit Wall Street.

While big banks favor much of the regulatory relief in the bill, they are wary of several of its provisions, including the changes made to the Federal Reserve and the requirement that they maintain high levels of capital to gain maximum regulatory relief.

"They oppose the bill," Hensarling said of big banks at Tuesday's hearing.

An analysis from the Congressional Budget Office found that none of the eight U.S. megabanks would likely choose the higher capital levels required by the measure to get out of all the new rules. In an update provided Tuesday, the budget office also found that the bill would save the government $34 billion over 10 years.

Two lawmakers, Republican Walter Jones of North Carolina and Democrat Marcy Kaptur of Ohio, sought a vote on an amendment to the CHOICE Act to re-impose Glass-Steagall, the Depression-era law separating commercial banks from investment banks and insurers. That amendment, which would have broken up the big banks if enacted, was rejected in the Rules Committee.