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House Speaker John Boehner's office rebuked Senate Democratic Leader Harry Reid Wednesday after he inserted a millionaire surtax to help pay for President Obama's $447 billion jobs bill, describing the change as a "desperate tax hike" gimmick.

The Democratic leader, who a day earlier turned away a crafty Republican bid to call a vote on the bill, dropped the surtax into the legislation in an effort to shore up support from Democrats who had voiced concerns with other tax provisions in the bill. Reid instead proposed a 5 percent income tax increase for people making more than $1 million a year.

Boehner spokesman Michael Steel said the surtax is not the way to go.

"Republicans have identified areas of common ground where we can work with the President and Democrats to create a better environment for job creation," Steel said in a statement. "That should be our focus, not desperate tax hike gimmicks floated to cover up divisions within the Democratic caucus."

But Reid defended the move, saying: "Even the Tea Party thinks it's time for millionaires and billionaires to pay their fair share."

The Nevada Democrat, without offering a date, pledged a vote on the bill "very, very soon."

Sen. Charles Schumer, D-N.Y., later added that the vote would be held "early next week."

The move comes after Reid turned away an effort by Senate Republican Leader Mitch McConnell to call up the bill for a vote Tuesday. Republicans were effectively trying to prove that President Obama, despite regularly calling on Congress to pass the bill, does not yet have enough support in his own party -- particularly in the Democrat-controlled Senate.

Reid decried McConnell's proposal as a political stunt.

On Wednesday, Reid said the package would be open to changes and that he "looks forward to amendments." Further, he guaranteed that passage of the jobs bill would keep unemployment below 9.1 percent.

However, House Republican Leader Eric Cantor said earlier this week that the GOP-controlled House will not accept the jobs bill as a complete package. Rather, he expressed interest in voting on specific provisions of it.

After three weeks of presidential demands for Congress to pass his jobs bill without delay, White House communications director Dan Pfeiffer said Obama was open to Reid's changes.

"We offered a balanced way to pay for the American Jobs Act, but if Congress has a better idea that ensures that everyone pays their fair share, we're open to it," Pfeiffer said.

White House Press Secretary Jay Carney also said Wednesday the administration is open to changes to how the bill is paid for. He said the priority is passage.

"The idea that our goal here is to use this as a political weapon, it's not. Our goal is to take action and put Americans back to work," he said.

Whatever changes are made to the revenue side of the ledger would, if the bill passes, factor into separate discussions being held by the bipartisan supercommittee trying to find at least $1.2 trillion in deficit savings over the next decade. Republicans apparently have opened the door to considering some revenue increases in exchange for benefit cuts. But the billions in tax hikes Democrats have proposed to pay for the jobs bill represent less than a third of the $1.5 trillion in total tax hikes Obama is calling for over the next decade -- that proposal is part of a more ambitious plan he presented to the deficit committee calling on the panel to go beyond its mandate. Republicans have indicated they will have a very limited tolerance for extensive revenue-raising tax changes.

While Republicans and Democrats appear to be pointing to a showdown that could reverberate into the 2012 election campaign, some elements of the measure could clear Congress with relative ease by year's end.

As an example, Republicans have not ruled out extending and expanding the payroll tax cuts that took effect on Jan. 1, at a cost of $247 billion over a decade, the single priciest item in Obama's legislation.

Democrats said Reid's proposed millionaires surtax was designed to quell much if not all of the opposition from his own rank and file, a subject that Sen. Dick Durbin of Illinois, the second-ranking leader, referred to in an interview with reporters.

The payroll tax cut extension faces little opposition from Democrats, and Obama's request for more than $100 billion in new spending as a way of creating jobs is also popular within his own party.

To pay for his $447 billion plan, Obama has proposed higher taxes on family incomes exceeding $250,000 and on the oil and gas industry.

The first request troubles Democratic senators from states like New York, New Jersey and California, where large numbers of families could be hit by the increase. The second has drawn opposition most prominently from Louisiana Sen. Mary Landrieu, whose state is home to numerous oil and gas operations.

The president also proposed higher taxes on hedge fund managers and corporate jet owners.

Reid predicted that by the time the jobs bill comes to the Senate floor, almost all Democrats would be behind it. "There could be -- I don't know who -- but there could be some that don't support it. But it would be a rare situation," he added.

In a speech in Texas, Obama referred to Cantor one day after the Virginia Republican said the White House's "all or nothing approach is unreasonable."

"Eric Cantor said that right now, he won't even let this jobs bill have a vote in the House of Representatives. That's what he said. Won't even let it be debated," the president said.

"At least put this jobs bill up for a vote so that the entire country knows exactly where members of Congress stand," Obama said. "Put your cards on the table."

Cantor's spokesman rejected the criticism.

"If House Republicans sent our plan for America's job creators to the president, would he promise not to veto it in its entirety? Would he travel district to district and explain why he'd block such common-sense ideas to create jobs?" Brad Dayspring said. "House Republicans have different ideas on how to grow the economy and create jobs, but that shouldn't prevent us from trying to find areas of common ground with the president."

House Republicans have begun passing legislation to block or roll back administration regulations on several industries, saying their removal will create jobs.

While Republican lawmakers appear receptive to tax cuts the president has called for, they have expressed strong opposition to his proposed new spending.

Fox News' John Brandt and The Associated Press contributed to this report.