The bill will retain significant cuts to spending on Medicaid, the government-run health coverage program for the poor. That is certain to be met with dismay by a number of moderate Republicans who opposed the first version of the bill for that reason.
The revised legislation will retain a set of Obamacare taxes, including a 3.8 percent surcharge on high-income earners.
The bill also would add an additional $45 billion toward tackling the opioid abuse epidemic, with that funding earmarked for substance abuse treatment and recovery.
There are just 52 Republican senators, which means that GOP leaders can afford to have just two members of their party's caucus oppose the bill if they hope to pass it. Vice President Mike Pence would break any tie in the favor of his fellow Republicans.
But at least eight GOP senators opposed the original version of the Senate bill.
Sen. Lindsey Graham, R-S.C., told reporters that the new bill was "well-received" by GOP senators when they heard about it from Majority Leader Mitch McConnell, R-Ky. in a meeting Thursday.
McConnell spoke on the Senate floor after that meeting.
He said that Americans were losing choices in their insurance plans under Obamacare, saying the current health-care law is headed toward "total meltdown."
McConnell said the Republican bill was made "stronger" in recent weeks with its revisions.
The new version was crafted after Senate GOP leaders, stung by an unfavorable analysis of the original bill, postponed a planned vote on that earlier version until after the July Fourth congressional recess.
The new version will be analyzed by the nonpartisan Congressional Budget Office.
The CBO's "score" of the revised bill, which is expected to be released early next week, will estimate how many more Americans would lack health insurance if the bill is signed into law, and also would project how insurance plan premiums would be affected.
In its analysis of the first version of the bill, the CBO estimated that it "would increase the number of people who are uninsured by 22 million in 2026 relative to the number under current [Affordable Care Act] law." That was only 1 million fewer than what the CBO had estimated a House bill would lead to.
"By 2026, an estimated 49 million people would be uninsured, compared with 28 million who would lack insurance that year under current law," the CBO said it its report on the earlier bill, released June 26.
It is by no means clear that the new CBO score will show significantly fewer people becoming uninsured under the new version of the bill than under the original draft of it.
A sign of the difficulties faced by GOP leaders came Wednesday from Sen. Rand Paul, R-Ky, who told reporters that day, before seeing the new version, that, "As far as I can tell, the new bill is the same old bill."
"I can't support it," he said.
Paul, like other more-conservative members of the GOP caucus, wants a bill that is more aggressive in repealing key elements of Obamacare.
In addition to Paul, moderate GOP senators like Dean Heller of Nevada and Susan Collins of Maine opposed the original Senate bill.
However, President Donald Trump on Wednesday told the Christian Broadcasting Network that he will be "very angry" if Congress does not send him a bill to replace Obamacare.
"I'm sitting waiting for that bill to come to my desk," Trump said. I hope that they do it. They've been promising it for years."
Craig Garthwaite, co-director of the Health Enterprise Management Program a Northwestern University's Kellogg School of Management, told CNBC that "broadly speaking, the legislation has some interesting new elements."
"Being able to pay premiums with pre-tax dollars is a good improvement, as is the change" funding to funding for hospital that serve a large number of Medicaid and uninsured individuals," Garthwaite said.
"That being said, without changes in the subsidy structure or in the Medicaid program the bill still amounts to a fundamental cut in access for low-income enrollees," Garthwaite said. "In addition, the attempt to create a set of non-compliant [health] plans invites adverse selection in a way that would be damaging, rather than stabilizing, to the individual market."
Adverse selection refers to the phenomenon when a disproportionate number of people with health problems enroll in an insurance plan, causing the plan to raise its premiums, which in turn discourages health customers from enrolling.