The White House says a new plan put together by GOP members in the house is "partisan attempt to appease a small group of Tea Party Republicans."
A possible "bright day" in the U .S. Senate could mean a long day in the House.
Senate leaders from both parties say a deal is near to end the partial government shutdown and avoid a potential default as soon as this week.
However, House Republicans decided Tuesday to offer their own proposal that would tack on provisions changing President Barack Obama's signature health care reforms in what appeared to be a last-gasp effort to influence the agreement to reopen the government and raise the federal borrowing limit.
According to GOP sources and confirmed by Republican Rep. Darrell Issa of California, the House plan would include most of what is in the Senate agreement while adding a provision to suspend an Obamacare tax on medical devices for two years and remove federal health care subsidies for Obama and legislators when they obtain health coverage under the reforms.
In addition, the House proposal would forbid the Treasury from taking what it calls extraordinary measures to prevent the government from defaulting as cash runs low, in effect requiring hard deadlines to extend the federal debt ceiling.
Issa said a House vote could occur as soon as Tuesday night, while Senate Majority Leader Harry Reid indicated his talks with GOP counterpart Mitch McConnell were moving toward an agreement.
"I am confident we will be able to reach a comprehensive agreement this week," Reid said Tuesday, reiterating the optimism he expressed Monday night that raised hopes among investors, world leaders and regular Americans that the shutdown stalemate was nearing an end.
"Perhaps tomorrow will be a bright day," he said Monday.
U.S. stock futures -- often seen as an indicator of how markets will open -- were up only slightly early Tuesday as investors largely waited out the politicians.
Obama spoke Monday with McConnell, according to a GOP Senate aide, but the White House canceled a planned meeting with congressional leaders in what was perceived as a move to give Reid and McConnell room to negotiate.
Sen. Bob Corker, R-Tennessee, said it was time to get a deal done after lengthy delays he blamed on the unrealistic goal of gutting Obama's signature health care reform law.
"The fact is we've got to figure out a way to move ahead," he told CNN's "New Day" on Tuesday. "In fairness, on our side of the aisle, we've wasted two months, focused on something that was never going to happen."
However, the House GOP decision to offer a counter-proposal promised further delay toward final congressional action on an agreement.
House Democrats immediately criticized the GOP plan, with Rep. Xavier Becerra of California calling it a "reckless attempt to try to circumvent what the Senate is doing" so close to the deadline for a possible U.S. default.
"That to me seems very irresponsible, and it certainly falls far short of being common sense," he said.
The negotiations are also being closely watched by other nations, which would also feel the impact should the United States run out of money to pay some of its bills.
Jon Cunliffe, who will become the deputy governor of the Bank of England, told British lawmakers over the weekend that banks should begin planning for contingencies.
The partial shutdown -- now in its 15th day -- has proved costly. Hundreds of thousands of federal employees are either idle at home or not being paid for their work during the shutdown. Government offices, many parks and other facilities are closed. And officials warn that tough choices are ahead about which bills to pay and which to let slide, should the shutdown and debt ceiling debate drag on.
So far, the standoff has cost the economy about $20 billion in gross domestic product, CNN's Christine Romans reported Tuesday on "New Day," citing Mark Zandi of Moody's Analytics. GDP is a measure of the goods and services produced by an economy.
At a visit Monday to a local food pantry, Obama warned of what he called continued partisan brinkmanship by House Republicans who "continue to think that somehow they can extract concessions by keeping the government shut down or by threatening default."
"My hope is a spirit of cooperation will move us forward over the next few hours," he said.
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