Formal announcement of the agreement will come at 12 noon ET on the Senate floor, a Republican Senate aide told Bash.
Republican leaders convened the Senate's full GOP caucus in the morning, and Sen. Kelly Ayotte of New Hampshire said on her way in that the announcement would be coming.
"I understand that they've come to an agreement but I'm going to let the leader announce that," Ayotte said.
Exact details of the Senate plan were not known. Nor was it clear how the Senate and House would proceed in considering the measure.
Both chambers would have to take special steps to get the legislation passed and to President Barack Obama's desk before the government's ability to borrow money expires on Thursday.
Legislators dropped hints on their way home on Tuesday that Senate Majority Leader Harry Reid and his Republican counterpart, Mitch McConnell, would quickly finalize an agreement in the works all week.
U.S. stocks opened sharply higher on expectations Washington would end its partisan fiscal impasse. The benchmark Dow Jones Industrial Average jumped 200 points.
According to sources, the Senate deal under discussion would reopen the government, funding it until January 15. It would also raise the debt limit until February 7 to avert a possible default on U.S. debt obligations for the first time.
It also would set up budget negotiations between the House and Senate for a long-term spending plan, and would include a provision to strengthen verification measures for people seeking government subsidies under Obama's signature health care reforms.
The focus shifted to the Senate after House Republicans failed on Tuesday to come up with a plan their majority could support, stymied again by demands from tea party conservatives for outcomes unacceptable to Obama and Senate Democrats, as well as some fellow Republicans.
It remained unclear if the congressional tea party wing led by Sen. Ted Cruz of Texas would continue efforts to force its demands into a congressional deal, perhaps by trying to filibuster or otherwise delay Senate action.
"It's up to him. I would hope he wouldn't," Ayotte, who represents New Hampshire, told CNN's "New Day." "Senators can cause to you run out the clock, but what's he trying to gain at this point? I would hope that whatever comes forward, that we would allow a vote on it as soon as possible."
Cruz, despite being in the Senate, is credited with spearheading the House Republican effort to attach amendments that would dismantle or defund the health care reforms known as Obamacare to previous proposals intended to end the shutdown.
All were rejected by the Democratic-led Senate, and Obama also pledged to veto them, meaning there was no chance they ever would have succeeded.
Ayotte called the tactic of tying Obamacare to the shutdown legislation "an ill-conceived strategy from the beginning, not a winning strategy."
However, Republican Rep. Steve King of Iowa advocated continued brinksmanship to try to change Obamacare, which conservatives detest as a big-government overreach.
"If we're not willing to take a stand now, then when will we take this stand?" he told CNN's "New Day," adding that if "the conservative Republican plan had been implemented five years ago, say at the inception of what is now the Obama presidency, we would have far less debt and deficit."
Despite warnings by Obama and economists that a U.S. default would spike interest rates and could have catastrophic impacts at home and abroad, King said he's not too concerned if the government passes Thursday's deadline to raise the borrowing limit.
"It's just a date they picked on the calendar," he said, adding that the government will still be able to pay the interest on its debt. "I'm more concerned about market reaction than I am of default itself."
Thursday marks the day the Treasury Department will run out of special accounting maneuvers to keep the nation under the legal borrowing limit. From that point on, it will have to pay the country's incoming bills and other legal obligations with an estimated $30 billion in cash, plus whatever daily revenue comes in.
The expectation is that the Treasury will be able to pay bills in full for a short time after Thursday, but exactly how long remains unclear. According to the best outside estimates, the first day the government will run short of cash could come between October 22 and November 1.
Officials warn that an unknown is whether creditors such as foreign countries that traditionally roll over their U.S. bond holdings could decide to instead cash out, creating a potentially major payout that the government would lack funds to fulfill.
What happens next?
A top GOP Senate aide said Wednesday that leaders in that chamber remain "optimistic an agreement can be reached," the same tone sounded Tuesday after lawmakers called it a night around 10 p.m. Senate staffers burned midnight oil to draft a framework bill.