Boehner faced criticism from some conservatives in advance of an expected vote on Wednesday.
The bill would raise the debt limit by $1 trillion while making cuts to federal spending of $1.2 trillion -- reductions that conservatives say aren't enough.
The measure also would establish a committee of lawmakers to recommend additional budget savings of $1.8 trillion, which would trigger an additional $1.6 trillion increase in the debt limit.
The White House objects to the requirement for a second vote before the 2012 elections.
Majority Leader Harry Reid said the measure stood no chance of passing the Senate even if it clears the House. He pronounced it "dead on arrival."
Washington and the nation are staring down an Aug. 2 deadline to raise the debt limit or face national default.
Flanked by conservative colleagues, Rep. Jim Jordan, R-Ohio, told reporters he could not back the Boehner proposal and said it doesn't have the votes to pass in the Republican-controlled House. In a two-step plan, Boehner is pressing for a vote on Wednesday and a second vote Thursday on a balanced-budget amendment to the Constitution.
"We think there are real problems with this plan," said Jordan, who heads the Republican Study Group. He argued that the spending cuts are insufficient and expressed opposition to likely tax increases.
Added Rep. Steve Southerland, R-Fla.: "If I had to vote right now, my vote would be no."
The conservative challenge came just hours after House Majority Leader Eric Cantor told the Republican rank and file to stop grumbling as he sought to rally lawmakers for the Boehner plan. In a closed-door session, the Virginia Republican acknowledged the resistance to increasing the nation's borrowing authority. "The debt limit vote sucks," he told his caucus. But Cantor insisted that it must be done.
Cantor spelled out the options for the GOP -- allowing default and stepping into an economic abyss, backing the Senate Democratic plan or calling President Barack Obama's bluff by backing the GOP's own proposal.
Neither of the rival plans offered by Boehner in the House and Reid in the Senate seemed to have the necessary votes in Congress amid a bitter stalemate that could have far-reaching repercussions for the fragile U.S. economy as well as global markets. Stocks fell Tuesday as U.S. markets registered their nervousness over the Washington gridlock between Obama and Republicans.
At the White House, spokesman Jay Carney said the administration remains in contact with congressional leaders despite the collapse of talks last Friday and inconclusive discussions this past weekend.
"We're working on Plan B. ... There has to be a product that can pass the House and the Senate and be signed into law," said Carney, who argued that the Boehner plan had no chance of passing in the Senate.
Carney insisted that Aug. 2 is the drop-dead date for the Treasury's cash flow -- "beyond that date we lose our capacity to borrow" -- and expressed confidence that the debt ceiling would be raised by the deadline.
The continued bickering on Capitol Hill overshadowed any signs of emerging common ground.
An influential conservative advocacy group urged lawmakers to vote against both plans. "Speaker Boehner's most recent proposal to raise the debt limit is regrettably insufficient to our times," Heritage Action for America said in a letter Tuesday to House members.
Still, Boehner told reporters that the "we are going to have some work to do to get it passed, but I think we can do it."
In the Senate, Reid challenged Republicans to back his competing legislation, arguing that the no-taxes, government-cuts proposal was just what they wanted.
"In short, it's everything the Republicans have demanded wrapped up in a bow and delivered to their door," Reid said at the start of the Senate session.
In a prime-time address Monday night, Obama pleaded for compromise and urged Americans to contact their lawmakers.
"We can't allow the American people to become collateral damage to Washington's political warfare," Obama told the nation.
Boehner, in a rebuttal, said he had given "my all" to work out a deal with Obama.
"The president would not take yes for an answer," he said.
Congressional officials said the House switchboard was near capacity with a high volume of calls and suggested using backup numbers. Websites also experienced heavy traffic, and lawmakers were sending out appeals for patience. A note from Rep. Doug Lamborn, R-Colo., simply said, "House Systems Maxed Out."
Unclear was whether the callers echoed Obama's argument or backed Boehner's call for his approach.
In afternoon trading on Wall Street, stocks were mixed as the financial markets warily watched the standoff.
The extraordinary back-to-back appeals by Obama and Boehner gave no indication that weeks of brinkmanship and sputtering talks over long-term deficit reductions were on the verge of ending. With the deadline rapidly closing, Congress and the White House had limited options to avoid a potential government default that could send the already weak economy into a damaging swoon.
Even as hints of progress seemed hard to find, Gene Sperling, chairman of the president's Council of Economic Advisers, said this was no time to be engaging in "doomsday scenarios."
Obama reiterated his call for achieving lower deficits though spending cuts and new tax revenues. But in a notable retreat, he voiced support for a Senate Democratic plan that would reduce deficits by about $2.7 trillion over 10 years only with spending cuts, not with additional revenue.
The Senate plan, unveiled by Reid, and the proposal announced the same day by Boehner overlap in significant ways. Both identify about $1.2 trillion in spending cuts to the day-to-day operating budgets of government agencies, though Reid's proposal also counts an extra $1 trillion in savings from winding down wars in Iraq and Afghanistan. Both proposals would create a bipartisan congressional commission to identify further deficit reductions, especially in major health care programs such as Medicare and Medicaid.
The primary difference between the two is timing. Reid's proposal would raise the debt ceiling enough so that it wouldn't have to be reconsidered until 2013, beyond the 2012 elections, as demanded by Obama. The GOP plan would only extend the debt ceiling for about six months.
For Republicans, the timing provides crucial leverage to force Democrats and the president to cut spending in Medicare, Medicaid and Social Security, expensive benefit programs that Democrats have long protected, despite escalating costs.
Credit rating agencies such as Moody's and Standard & Poor's have threatened to downgrade the United States' gold-plated AAA rating if Congress and the White House don't extend the debt ceiling and take steps to bring long-term deficits under control.
While both plans would increase the debt ceiling, ratings agencies have said a short-term increase such as the one proposed by House Republicans may not be enough to protect the U.S. from a ratings downgrade. What's more, neither plan offers the larger deficit-reducing assurances that credit ratings have said they need for the U.S. to retain its place as one of the most secure investments in the world.
Sperling was interviewed on MSNBC.