Story highlights Thinking is no deal on shutdown until there is a plan for a bigger agreement with debt deadline looming

Boehner faces splintered GOP caucus on what demands should be attached to debt ceiling plan

He has told fellow GOP lawmakers he might seek help from Democrats, if necessary, to avoid default

President Barack Obama has said he would not negotiate on debt ceiling, calls for vote to end shutdown

Many House Republicans concede the government shutdown likely won't be resolved until the next major congressional deadline in mid-October when Congress needs to raise the nation's debt ceiling.

"I don't see anything happening with the shutdown until we get to some kind of a plan for a bigger agreement," Louisiana Rep. Charles Boustany told reporters after House Speaker John Boehner and other GOP leaders huddled with House Republicans on Friday.

Like the fight over the government spending bill, Boehner faces a splintered GOP conference on what kinds of demands should be attached to legislation raising the debt ceiling.

Boehner told fellow GOP legislators earlier this week that he doesn't want the United States to default on its debt and would try to prevent that even if it meant getting help from Democrats to pass the necessary legislation.

This means he sets aside a practice conservatives insist he follow known as the "Hastert rule," according to a Republican House member who requested anonymity to talk about the private meeting.

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The "Hastert rule" adopted by former GOP House Speaker Dennis Hastert, is the practice of not permitting a vote on any legislation in the House unless a majority of House Republicans back it.

Boehner never pledged to follow the rule when he assumed his post, and has been criticized by some House conservatives and several outside groups for those instances when he allowed votes on measures that passed with more Democrats than Republicans.

But Boehner is adamant that he will not do what Democrats and President Barack Obama want - take up a "clean" increase in the debt limit -- a point he reinforced to reporters on Friday.

"I think the American people expect if we're going to raise the amount of money we can borrow we ought to do something about our spending and the lack of economic growth in our country," Boehner said at a news conference.

Congressional Democrats repeat, as does the president, that they will not negotiate over the debt ceiling.

But the last time Congress wrangled with this issue in 2011 Obama did negotiate and at the last minute Congress passed legislation that contained some major deficit reduction measures as part of the debt limit increase.

Even though the debt deadline is less than two weeks away, top House GOP leaders didn't discuss on Friday how they plan to address it.

Instead, multiple GOP members leaving that session said it focused primarily on the current strategy to keep passing piecemeal bills that fund parts of government and insist that Democrats need to sit down and negotiate over a government funding bill.

But there are smaller group discussions about how to address the debt limit. The speaker met Thursday with a small circle of GOP members with whom he regularly consults to get a pulse of what House Republicans want as part of a deal to increase the Treasury's borrowing authority.

"Boehner told us that we've got to get something with regard to budget and the debt in order to raise the debt ceiling" one GOP member who attended the session told CNN.

Boustany doesn't like the term "grand bargain" -- the name given the deal that eluded Boehner and the President in 2011, but he thinks House GOP members could get behind what he called a "substantial budget agreement" as part of the debt ceiling talks, though he's concerned about "the short time frame."

Boustany outlined what he thinks could serve as a framework -- some of the changes to entitlement programs that Obama has suggested before and some detailed targets for tax reform, both items that Republicans want.

He believes in return GOP members could agree to raise the debt limit potentially through the rest of Obama's term.

Mentioning that Democrats have pushed to eliminate the forced spending cuts that came out of the last budget deal, Boustany said "turning off sequestration is something we could put on the table in exchange for other savings. We're looking for a real debt management approach."

But the House Republican who met with Boehner said given the calendar there may not be the time to get big policy changes in place for the things that Republicans would agree to and also get some Democrats on board.

"Everyone understands that the Obamacare issue may be out of reach, probably always was, but if the president thinks he is going to be able to hold ground with negotiations on the debt limit he sounds like Ted Cruz," that lawmaker said.

Multiple House GOP aides told CNN that despite the meetings with Boehner and others on the debt ceiling, there's still no consensus on what conditions they'd attach to a debt increase.