Updated 12:36 p.m. | House Republican leaders are waiting to hear back from the White House after GOP staff sent over an informal proposal Thursday night that would raise the debt ceiling and create a process for reopening the government.

Aides say President Barack Obama and Republicans are haggling over a deal that would lead to an end to the two-week-old government shutdown and raise the debt ceiling in exchange for a clear path to serious negotiations about spending and other fiscal issues. But negotiations are fragile, they say.

“It’s my understanding that there was a sort of a final — not offer, but, you know, ‘OK, here’s our position,’ that was left with the president’s people somewhere around 10 or 11 o’clock last night,” said Rep. Tom Cole, R-Okla., a close ally of Speaker John A. Boehner, R-Ohio.

Cole said Republicans are waiting to hear back, but “there’s not much sense to moving ahead if they’re not willing to sit down and talk.”

It appears Republicans are asking for a path to a broader fiscal deal for reopening the government and raising the debt limit. The impetus for reaching the deal would still be a deadline on the continuing resolution and the debt limit, which would only be short-term.

The Associated Press reported Friday that the proposal House Republicans sent Obama includes ending the shutdown, increasing the debt limit and easing automatic spending cuts known as the sequester in exchange for cutting entitlement programs. It would include raising the cost of Medicare for wealthier individuals, the AP reported.

While the White House and Republicans are likely going back and forth over the length of the agreement and the conditions for negotiations on a broader fiscal deal, many Republicans are expected to still vote against such a deal.

Still, GOP leaders will likely need some Democratic votes to get passage of the short-term measure or measures.

“Once the two sides have an agreement, then an agreement means there’s votes from both sides. The president has already signaled, ‘Look, I’m willing to sign short-term extensions, do things like that,’ I think that was a move in the right direction as well on his part,” Cole said. “So it looks to me like the two sides have inched a little closer to one another in the last 24 hours, and I’d like them to step closer instead of inch closer, but at least it’s moving in the right direction.”

He added, “I’m not sure that we would need them, honestly, I think there’s a pretty strong Republican unanimity on this, but, you know, if it’s an offer that’s acceptable to the president, then Democrats might well want to join in as well.”

Rep. Virginia Foxx, R-N.C., said the Rules Committee would be meeting “some time today, I believe.”

Indeed, Rules announced a meeting Friday on moving the farm bill to conference with the Senate, but there was no legislation on the debt limit or the continuing resolution — yet.

Rep. Edward Whitfield, R-Ky., said Republicans seemed generally warm to the idea of a CR and debt-limit deal.

“Depending upon the sincerity of the effort to negotiate on some other things,” he said.

But other Republicans still seemed to not have received the memo — nor seen the recent poll numbers, which show Republicans with historically low approval ratings.

“Our position right now is no CR, until we get, well, really no CR, but to delay the debt limit for about five weeks in exchange for the White House sitting down and talking with us,” Rep. John Fleming, R-La., told reporters Friday afternoon.

That was the plan discussed in conference Thursday morning, and that was the offer Republicans took to the White House Thursday evening. But negotiations seemed to have progressed from that position.

Still, any deal will likely have to be announced — and approved — by the Republican conference before GOP leadership moves ahead, and that could slow down the negotiations. House GOP leaders have called a full conference meeting for 9 a.m. Saturday morning, where presumably they expect to propose a way forward.

Senate Republicans and Democrats are working on their own agreement to end the shutdown and raise the debt limit, but those talks appear to be largely separate from the House GOP’s talks with the White House.