U.S. House Speaker John Boehner says the FBI will have to answer questions about its 2011 investigation of bombing suspect Tamerlan Tsarnaev. Reuters

In their latest move in the "fiscal cliff" sparring, House Republicans announced Tuesday night that they will pass their own tax bill as a backup plan to avert the tax hikes and automatic budget cuts looming at New Year’s.

The “Plan B” bill, as Speaker John Boehner, R-Ohio, calls it, would extend the Bush tax rates cuts for everyone with an annual income under $1 million. Senate Majority Leader Harry Reid, D-Nev., has already declared it did in his chamber.

President Barack Obama's latest offer, announced Monday night, puts the threshold for income tax hikes at $400,000.

House Majority Leader Eric Cantor, R-Va., said he expects a vote on the Republican plan on Thursday. A simple majority will be enough to pass without any Democratic support.

While the move may not prompt Obama to give further ground in his negotiations with Boehner, it could let Republicans argue they did what they could to stop tax hikes and the full impact of the "fiscal cliff," which many economists warn could trigger another recession.

It remained unclear Tuesday, The Washington Post noted, whether Boehner was using the plan to extract more concessions from Obama or to secure more support from his caucus later for an eventual bargain with the White House.

"Why not put on the floor something that's what most Americans think the president is talking about, which is protecting from tax increases everybody but truly millionaires and billionaires?," said Rep. Pat Tiberi, R-Ohio, Reuters reported.

When it dies in the Senate, he said, "that's not our problem. We can't be held responsible for what the Senate does."

Polls have consistently suggested that the public is likely to blame Republicans for failure to reach a deal ahead of the Dec. 31 deadline.

In July, the Democratic-controlled Senate passed a bill to extend the current low rates for all families with net incomes below $250,000 a year. The House Republican proposal, if passed by the House, would require agreement by the Senate or force a round of negotiations on a compromise between the two chambers.

The White House seemed unmoved by the Republican tactic, but stressed Obama's willingness to compromise further.

"The president has demonstrated an obvious willingness to compromise and move more than halfway toward the Republicans," White House spokesman Jay Carney told reporters, adding that Obama is making a "good faith" effort to reach a compromise.

The president dispatched his top congressional negotiator, Rob Nabors, to Capitol Hill Tuesday to quell uprisings among House and Senate liberals upset that the White House had made a key concession by agreeing to apply a less generous calculation of inflation for federal programs, including Social Security, the Post reported.

Conservatives, meanwhile, voiced skepticism about Boehner’s “Plan B,” saying it would violate their principle of opposing any tax increases while also failing to spare the Pentagon from deep budget cuts starting next year.

At an evening meeting with House Republicans, Boehner made the case that his new plan would lock in tax cuts permanently for more than 99 percent of taxpayers, even if millionaires’ tax rates increased. The speaker said his approach would buy time so that he could reach a final deal with Obama early next year to address the remaining budget issues.

After the meeting, Rep. John Kline, R-Minn., told Bloomberg News he could not predict how much support Boehner would get for his plan because that’s “hard to quantify.”

Still, the discussion showed “there is an understanding” that “absent action we are going to be faced with a huge, huge tax hike,” Kline said.