House Republicans plan to work "throughout the night" Thursday to cobble together a deal with Democrats that would raise the nation's borrowing limit, but they also likely will have to agree to reopen the government or risk rejection by Democrats.

House Republicans left a White House meeting with President Obama late Thursday afternoon after receiving a tepid reception for a plan they offered: Raise the debt ceiling for six weeks in exchange for negotiations on major spending reforms.

Obama, according to those at the meeting, pressed Republicans to include in the deal federal funding needed to end a government shutdown that on Friday enters its 11th day.

"The president’s goal remains to ensure we pay the bills we’ve incurred, reopen the government and get back to the business of growing the economy, creating jobs and strengthening the middle class," said an administration statement issued about 90 minutes after the Republican leaders exited the White House through a side door away from reporters.

The GOP's original proposal, which House Speaker John Boehner first offered early Thursday, does not specifically address the shutdown. Republicans want to deal only with the debt ceiling, which they said they're willing to raise for six to eight weeks just to give them time to negotiate that government funding bill and spending reforms with Democrats.

But Obama insists Republicans approve the government funding bill first. And Senate Republicans may have provided him with the additional leverage he'll need to press that demand. The Senate GOP is crafting a deal of its own, one that both raises the debt ceiling and provides the money needed to reopen the government.

The Senate proposal would extend the debt ceiling increase and government funding for a year. It also would accomplish two goals sought by conservatives who wanted to defund Obamacare: Repeal of the 2..3 percent medical device imposed under the new health care law and require income verification for those who seek health insurance subsidies.

Sen. Susan Collins, R-Maine, is the leading author of the proposal and said she is optimistic it can pass the Democratic Senate.

"I see this as a real breakthrough," Collins said after a closed-door meeting with Senate Republicans.

"The important thing is," Collins added, "we have an obligation to govern and governing means getting the government reopened. and fixing some of the problems with Obamacare and dealing with the debt limit in a responsible way."

A GOP leadership aide said "versions" of the Collins plan "are under discussion."

Senate Republicans will present their plan to Obama in a White House meeting Friday.

Senate Democrats, meanwhile, had their own meeting with Obama earlier Thursday and left the White House insisting that any deal would have to reopen the government.

"Not going to happen," Majority Leader Harry Reid, D-Nev., said, when asked after the meeting if he would negotiate with Republicans before funding the government's 2014 fiscal budget.

Reid wants to extend the debt limit until after the 2014 elections. He will call a vote on his own debt ceiling increase, a bill that raises the nation's borrowing limit until December 2014 but includes none of the policy add-ons Republicans have demanded.

Republican House leaders returned to the Capitol late Thursday following their meeting with Obama and the 19 lawmakers gathered in Boehner's office to plot their next move.

Majority Leader Eric Cantor, R-Va., said the White House meeting "was clarifying for both sides as to where we are."

House Appropriations Committee Chairman Hal Rogers, R-Ky., made one thing clear: Any final deal is going to have to address both the temporary government spending bill, known as a continuing resolution or CR, and the debt ceiling.

"We're trying to find if there is a way to quickly settle the CR questions so that we can pass a CR and stop the shutdown," Rogers told the Washington Examiner. "I think it's clear [Obama] would like to have the shutdown stopped, and that would require a CR."

"We're trying to find out what he would insist upon in a CR, and what we would insist upon, Rogers said. "So, our staffs are meeting and we'll talk later tonight."

Correspondent Sean Lengell contributed to this report