The deal would also give states flexibility on what kinds of health plans could be sold in individual states, and would reportedly allow people over age 30 to buy less expensive and less comprehensive catastrophic health insurance plans. The deal also would allocate money for states to promote enrollment in health coverage.

Alexander said the bipartisan deal — which would have to be approved by Congress — would reinstate the cost-sharing reduction reimbursement payments to Obamacare plans for two years. "This agreement avoids chaos," Alexander said on the Senate floor. "I don't know a Republican or Democrat who benefits from chaos."

Trump's move, coupled with an executive order he signed Thursday, is widely expected to harm the Obamacare markets. It was already leading to higher premium prices for individual health plans next year.

A Republican senator said he has reached a tentative deal with a Democratic counterpart to provide a short-term stabilization fix to Obamacare health insurance markets, NBC News reported.

And it would allow for the creation of "interstate compacts" for the purchase of health coverage.

There is no guarantee that Alexander and Murray, who have been discussing a possible deal for the past month, will be able to win approval for their agreement from either the rest of Congress, or from Trump.

"We are willing to work with Congress to reach a legislative solution," a White House official told CNBC.

"We will not provide bailouts to insurance companies until we provide the American people with relief from the Obamacare disaster," that official said.

Murray said senators are still "ironing out a few of the last details right now" about the bill.

Senate Minority Leader Chuck Schumer, D-N.Y., saying "I want to salute both Lamar Alexander and Patty Murray," called it "a good solution."

"It stabilizes the system," said Schumer. "There's a growing consensus that in the short term, we need stability in the markets."

Schumer said the bill includes "anti-sabotage" provisions to counteract the "sabotaging" of Obamacare that Trump has engaged in.

"This agreement would undo much of that sabotage," Schumer said.

Marc Short, White House director of legislative affairs, told NBC News that the deal in its current form is not close to being something that the Trump administration would accept.

Short said the administration would want much more than the flexibility being offered to individual states.

During a White House news conference Tuesday, Trump said, "We have been involved" in the Alexander-Murray deal. He added that it could "get us over the hump."

"This is a short-term deal, because we think, ultimately, block grants going to the states is going to be the answer," Trump said.

He was referring to the block grant system that was contained in a failed Obamacare replacement bill last month, that would have given states money directly to set up their own health coverage systems for individuals.

Trump said there currently is "this very dangerous little period, including [a] dangerous period for insurance companies."

Trump's repeated threats to end the CSR payments had already triggered higher insurance premium rates in a number of states for 2018 even before he followed through on those threats.

Caroline Pearson, a senior vice president at the Avalere Health consultancy, said she was "not particularly optimistic that some congressional action can save the day here."

"It's too late in terms of timing. Just because they've announced a deal — they don't have the votes," Pearson said. "They've had trouble producing the votes in the past."

"Open enrollment begins" Nov. 1, Pearson noted. "We're just sort of out of time. And the question is could we throw everything into a big scramble in December and try to get it sorted out in January."

Trump's decision last Thursday to end the reimbursement directly led to higher premium rates in some states, with more expected in coming days.

Pennsylvania insurance regulators on Monday approved Obamacare rates that on average will be 30.6 percent higher next year — with the biggest chunk of that boost due to Trump's move.

Obamacare customers in Pennsylvania had been on track to face rates that were about 7.6 percent higher on average before Trump ended the cost-sharing reduction reimbursement payments.

In Colorado, Obamacare customers are looking at rates that will be at least 6 percent higher than the original 27 percent hike because of the move.

And in Alaska, where rates were actually set to drop by more than 26 percent next year, the Trump CSR decision means that the actual drop in rates will end up being 20 to 22 percent, according to a leading Obamacare rate tracker.

One outlier so far is North Dakota, whose insurance commissioner on Tuesday said he would not let Obamacare insurers readjust their rates upward for 2018 despite Trump's move.

North Dakota recently had seen the insurer Medica drop out of the Obamacare market for next year because regulators would not let Medica submit proposed rates assuming the CSRs would stop being paid.

"There are a lot of people who are going to get hurt" by the president's decision, said Charles Gaba, who operates the Obamacare data site ACASignups.net.

"It just depends on their state, really," Gaba said.

He said his projections show that, on average, Obamacare plans of all types nationally will cost about 14 percent more than they otherwise would have because of the ending of the CSR payments.

With the CSR cut assumed, Gaba said, rates will be 30 percent higher on average. Without it, they would have been 15 or 16 percent higher, he said.

However, as Gaba noted, the actual price hikes paid by customers will depend on where they live, in large part.

Most Obamacare customers who receive subsidies that reduce their monthly premiums will be insulated from the price hikes, either largely or completely, because those subsidies rise as premiums rise.

However, nonsubsidized customers face the full brunt of the price hikes.

Some states have taken steps to protect those customers by limiting the CSR-related price increases only to so-called silver plans — which are the only type of plans that offer CSR discounts. Other states have limited the CSR-related price increases only to plans sold on Obamacare exchanges — plans sold outside of those marketplaces do not have the CSR-affected prices.

Negotiations by Alexander and Murray were jump-started last Thursday when the Trump administration said it would immediately cease paying the CSRs.

The reimbursements are meant to compensate insurers for discounts that they must — by law — offer to low-income Obamacare customers for out-of-pocket health costs, including copayments, coinsurance and deductibles.

Trump had threated for months to kill the resimbursements to insurers, which would have left them holding the bag financially for the cost of those discounts. On Thursday, his administration made good on those threats after Attorney General Jeff Sessions said the payments were illegal, given the lack of explicit appropriation of them by Congress.

The reimbursements would have been worth about $10 billion to insurers next year.

Avalere Health on Tuesday estimated that insurers will lose $1 billion in CSR payments through the end of 2017 as a result of Trump's decision.

Because of Trump's threats, a number of insurers, such as North Carolina's Blue Cross and Blue Shield, had asked for premium price increases next year that were as much as 20 percent or so higher than they otherwise have been. BCBC later dropped that request to 14.1 percent — but 5.3 percentage points of that was still due to uncertainty over the CSRs.

Other insurers, notably Anthem, dropped out of a number of states because of uncertainty over the CSR payments.

Some states told insurers to file two sets of proposed rates: one assuming the CSRs continued being paid, the other assuming they would be cut off.

California was one of those states.

And last week, even before Trump's decision to end the CSRs, California's Obamacare market imposed a 12.4 percent surcharge on its most popular type of individual health plans to cushion insurers for the possibility of the payments being ended.

California is one of 14 states that, along with the District of Columbia, operates their own Obamacare marketplace.

People in the rest of the country can buy Obamacare plans on HealthCare.gov, the federally run insurance exchange.

Caitlin Oakley, a spokeswoman for the U.S. Health and Human Services Department, said, "Insurers in the vast majority of states on the federal exchange submitted rates for the upcoming plan year assuming that CSR payments would not be made, so no rate adjustment is needed."

Oakley said the federal Centers for Medicare and Medicaid Services, which oversees HealthCare.gov, "is working on a case-by-case basis with those states where regulators explicitly required insurers to assume CSR payments would be made."

Lori Lodes, a former top HHS official, told CNBC on Tuesday that the series of actions by the Trump administration to undercut the Obamacare markets has led to widespread confusion among Obamacare customers and potential customers.

Lodes said some people believe that Obamacare, which requires most Americans to have some form of health coverage or pay a fine, is no longer the law of the land.

Others mistakenly believe that cost-sharing discounts to customers have ended — when in fact only the reimbursements to insurers who offer those discounts are going away, she said.

Because of this confusion, Lodes said, "there is zero question that it's going to depress enrollment" in Obamacare plans in 2018.

"The question is, can we do enough to limit the damage from it," she said.

Lodes and other former Obama administration officials recently launched a group, Get America Covered, to counteract what they have called sabotage of the health-care law by the Trump administration.

On Tuesday, Get America Covered distributed a poll from Hart Associates that underscored the widespread confusion about health insurance options that persists among many Americans.

The poll found that just 31 percent of people who have Obamacare plans sold by a government-run marketplace know that open enrollment in plans for 2018 begins Nov. 1. Just 12 percent of uninsured people are aware of that date.

Only about half of the insured customers were aware of the fact that they qualify for a federal tax credit that lowers their monthly premiums, and just 15 percent of the uninsured respondents knew that, according to the poll.

"In fact, just 41% of individuals currently insured through the marketplace are aware that they currently receive a tax credit," according to a report by Hart Associates, which conducted the poll for the Service Employees International Union.

Lodes noted that the poll, which questioned 200 insured people and 200 uninsured people, was conducted a week before Trump made his decision to end the CSR payments.