By MJ Lee, Lauren Fox, Phil Mattingly and Tami Luhby, CNN

(CNN) -- As the Republican Party's last-ditch effort to repeal Obamacare hangs by a thread, a revised version of the Graham-Cassidy bill was circulated to Senate Republicans on Sunday with the aim of winning over key votes.

Even with the new changes, the task ahead is daunting. GOP Sens. Rand Paul and John McCain have already publicly opposed the bill, and leadership can't afford to lose one more. Plenty of others, including Lisa Murkowski of Alaska and Susan Collins of Maine have made known their deep reservations -- Collins went as far as to say on CNN Sunday morning that it was "very difficult" for her to envision getting to a "yes."

And time is ticking. The vehicle that Republicans are using that allows them to advance a bill without any Democratic support is set to expire at the end of September, making the next several days critical.

Senate Majority Leader Mitch McConnell will soon have to make a call: walk away from his party's latest attempt to gut the Affordable Care Act, or put Graham-Cassidy on the Senate floor and hope that it passes.

Neither option is ideal. GOP lawmakers are now once again at risk of confronting backlash from their base for failing to overhaul Obamacare, while another unsuccessful vote would deal a major blow to McConnell just two months after his first attempt to pass a repeal bill failed in dramatic fashion.

An internal GOP analysis, circulated to Senate offices, shows spending boosts states like Alaska and Kentucky -- data that will almost certainly be used to sell the revised proposal in the days ahead.

But notably, those increases in the projection incorporate savings from ending the state match of Medicaid expansion. So in total, there is still a reduction in health care spending in these states when compared to current law. (For example, Alaska would get a $100 billion cut via its block grant in the bill, but the GOP analysis shows it would receive a boost of three percent.) The analysis also doesn't address the overhaul of the Medicaid program, from an open-ended entitlement program to a per person cap.

How the new version helps Alaska

In an important nod to Murkowski, the revised bill says Native Americans and Alaska Natives enrolled in Medicaid expansion prior to 2020 could continue to be eligible after that point, according to documents circulated Sunday night to senior Senate aides and obtained by CNN. The state's sizeable native population and that group's unique health needs has been a serious concern for Murkowski.

In one new provision particularly beneficial to Alaska, the state would receive a 25% boost in federal matching funds for Medicaid due to its defined high-level of poverty.

Speaking of Medicaid expansion

The newest version of the bill gives Medicaid expansion states that expanded the program after 2015 additional federal dollars.

Under the latest proposal, states that expanded Medicaid after December 2015 would be eligible for $750 million that the federal government would allocate between 2023 and 2026. According to a summary of the bill, "the administrator will distribute fund to qualifying states based on their percentage of low-income individuals in the state compared with the total number of low-income individuals for all qualifying states."

States can design their own rules

The updated bill would also allow states to design some of their own insurance rules. This would wipe away many of Obamacare's protections not only for those with pre-existing conditions, but also for those who get medical care.

It would let states alter the essential health benefits, which require insurers to cover prescription drugs, mental health and other services. But states could also change many of Obamacare's financial protections, including limiting how much people must pay out of pocket each year and how much of the tab insurers must pick up. That means that insurers could once again offer plans with deductibles of $10,000 or more for a single person. Republicans have long criticized Obamacare for having high deductibles, which this year cannot exceed $7,150.

Also, states could allow insurers to create multiple risk pools -- meaning those who are healthy could be assigned to one risk pool with lower premiums, while those who are not could be put into another with higher rates.

The new bill would also let states change the Obamacare requirement that insurers could not charge older Americans in their 50s and early 60s more than three times the rates of younger enrollees. Under the Affordable Care Act, younger consumers essentially subsidized the premiums of older ones. All the GOP repeal bills this year have loosened this provision, which has led the influential AARP to blast Republicans for levying an "age tax" on older Americans.

And under Obamacare, many preventative services -- including annual check-ups, mammograms and colonoscopies -- were free. States could make changes to this provision too.

Loosening the insurance regulations has been a priority for Paul, whom Republicans are also trying to woo.

Pre-existing conditions protections have been a flashpoint

The new proposal also appears to attempt to shore up concerns about states adequately addressing pre-existing conditions if they opt out of certain Obamacare regulations. It appears the new plan would attempt to address that by saying states "shall" maintain access to adequate and affordable coverage, which is a shift from a state saying it "intends" to maintain them.

In one of the most notable changes from Obamacare, states would receive block grants from 2020 to 2026. In applying for such funding, states must describe how they would provide "access to adequate and affordable" coverage to people with pre-existing conditions -- a requirement that critics and experts have widely said would not ensure that people with medical histories would not see spikes in their coverage costs.

The issue became a flashpoint in recent days when late-night comic Jimmy Kimmel accused the authors of Graham-Cassidy of failing to preserve protections for people with pre-existing conditions, pointing to his own son who was born with a congenital heart defect.

As in the previous version of the bill, the new Graham-Cassidy would repeal the individual and employer mandates -- penalties imposed by the Affordable Care Act on people who don't have insurance and certain employers that don't provide coverage for employees.

This story has been updated and will continue to update with new information and analysis.