Boehner, the top Republican in Congress, had earlier conceded on Obama's insistence that tax rates rise on the wealthiest Americans, but the two have been unable to agree on what income levels should be included in that category.
Analysts said Obama and Boehner may strike a compromise at $500,000 or close to that, though time was running short.
One House Republican aide, asked about prospects for "Plan B" on the House floor, said: "It wouldn't be surprising ... if a lot of conservatives balk at something like that." The House's second ranking Republican, Eric Cantor, said he was confident his party members in the House would back the bill.
'WE CAN DO BETTER'
Even as he presented the measure, Boehner said he would continue to negotiate with Obama on a broader agreement.
"Plan B is Plan B for a reason. It's a less-than-ideal outcome. I've always believed we can do better," Boehner said.
The expiration of low tax rates enacted under former President George W. Bush is a key component of the "fiscal cliff" that lawmakers are trying to prevent from taking hold next month, along with deep automatic government spending cuts.
Often challenged by the conservative wing of his caucus, Boehner held Republican lawmakers together in support of his efforts to forge a deal with Obama. The speaker emerged largely unscathed from a potentially tough meeting with his fellow House Republicans on Tuesday morning.
Representative Darrell Issa, a key committee chairman, said his fellow House Republicans "were supportive of the speaker. ... I saw no one there get up and say, 'I can't support the speaker.'"
With opinion polls showing broad support in the United States for raising taxes on the wealthiest Americans and Obama still buoyed by his re-election last month, the Republicans' traditional opposition to tax hikes has waned somewhat.
The Obama-Boehner talks have largely overcome stark ideological differences and are focused increasingly on narrower disagreements over numbers.
Obama also may face unrest from within his party. Liberal Democrats were likely to oppose a key compromise he has offered to permit shrinking cost-of-living increases for all but the most vulnerable beneficiaries of the Social Security retirement program. His proposal calls for using a different formula, known as "chained Consumer Price Index," to determine the regular cost-of-living increases, essentially reducing benefits.
"I am committed to standing against any benefit cuts to programs Americans rely on, and tying Social Security benefits to chained CPI is a benefit cut," Democratic Representative Keith Ellison said in a statement.