The proposal, which the Ohio Republican dubbed "Plan B," would leave in place military and domestic spending cuts that Republicans have said could have horrible consequences, especially to national defense.
Lawmakers from both parties gave the proposal little chance of passing. Some told The Wall Street Journal they saw the offer as a largely tactical maneuver on Boehner's part.
The White House immediately rejected "Plan B," with spokesman Jay Carney saying it could not pass the Senate and "therefore will not protect middle-class families" from large tax increases scheduled to start Jan. 1.
"The president has put a balanced, reasonable proposal on the table that achieves significant deficit reduction and reflects real compromise by meeting the Republicans halfway on revenue and more than halfway on spending from where each side started," Carney said in a statement.
President Barack Obama will not accept a deal "that doesn't ask enough of the very wealthiest in taxes and instead shifts the burden to the middle class and seniors," Carney said.
Speaking with reporters at the White House Wednesday, Obama said he and Boehner are close to agreement, and urged Republicans to "take the deal."
"What separates us is probably a few hundred-billion dollars," Obama said. "The idea that we would put our economy at risk because we can't bridge that gap doesn't make a lot of sense."
Obama said the White House has met congressional Republicans "at least halfway in order to get something done for the country" and Republicans were in a position until a few days ago to say they got "a fair deal."
Afterward, Boehner spoke briefly with reporters on Capitol Hill
"I hope the president will get serious soon about providing and working with us on a balanced approach," he said. "Then the president will have the decision to make: He can call on Senate Democrats to pass that bill or he can be responsible for the largest tax increase in American history."
White House Communications Director Dan Pfeiffer said Boehner's plan amounts to a tax break of $50,000 for millionaires while eliminating tax credits for student loans and other benefits important to the middle class.
"The president urges the Republican leadership to work with us to resolve remaining differences and find a reasonable solution to this situation today instead of engaging in political exercises that increase the possibility that taxes go up on every American," Pfeiffer said in a statement.
Obama Monday offered a plan that would allow current income tax rates to expire on income over $400,000 -- a change from his earlier insistence that the increase apply to income over $250,000. Obama's offer would produce $1.2 trillion in tax increases and cut $930 billion in spending over 10 years.
Boehner called Obama's plan unbalanced and inadequate.
"What we've offered meets the definition of a balanced approach, but the president is not there yet," Boehner said Tuesday.
He said he was not giving up on talks with Obama. But with more than $500 billion in tax hikes and across-the-board spending cuts set to take effect in 13 days, he had to come up with a fallback plan to make sure most Americans don't see their taxes rise.
Republican leaders tentatively planned a House vote Thursday on Boehner's proposal. But that vote would follow a vote on a Senate-passed bill to extend the expiring George W. Bush-era tax cuts for incomes below $250,000, The New York Times said. That vote is expected to fail, as a show to Obama his initial $250,000 offer could not pass.
Some Republicans said they would be hard pressed to support any plan -- including Boehner's backup option -- that raises tax rates, even if only on income above $1 million.
House Armed Services Committee Chairman Howard McKeon, R-Calif., said he did not think he could support Boehner's bill because it would cut $500 billion from the military over 10 years.
Rep. John Fleming, R-La., told the Times: "Why go on record raising taxes on anybody if it won't cut spending and won't even become law? I haven't found a way of supporting that."
Other GOP lawmakers said they might grudgingly accept a tax increase on millionaires.
"I hate it. I hate it," Rep. Jason Chaffetz, R-Utah, told the Journal. "But I'm trying to be reasonable."
Despite the handwringing, Sen. Chuck Schumer, D-N.Y., said he expected Boehner and Obama would work out a deal that could pass both chambers.
"They seem to be so close that I'd be surprised if it fell apart," Schumer, a member of the Senate Democratic leadership team, told reporters on a conference call. "I think the likelihood is we will get an agreement."