Washington (CNN) President Donald Trump would sign the Graham-Cassidy bill if the legislation to repeal Obamacare makes it to his desk, an administration official told CNN Tuesday, as the White House launched a full-court press on Capitol Hill.

The White House has been working with Republican Sens. Lindsey Graham of South Carolina and Bill Cassidy of Louisiana for weeks on the new effort. A second source added that Trump himself had quietly encouraged the two senators to take another shot at health care.

Vice President Mike Pence and other administration officials are on Capitol Hill Tuesday to lobby Republican senators on the bill, which faces a September 30 deadline to pass the Senate with a simple majority. Republicans can only lose two senators or the effort will fail.

Once again, the key votes are Sens. Lisa Murkowski, Susan Collins, Rand Paul and John McCain, who have been surrounded by reporters in the Capitol for the past two days.

The President has made calls to senators in recent days in an effort to urge them to move forward with health care one more time. In these calls, Trump has warned Republican lawmakers that this could be their last chance to follow through on a seven-year promise to repeal President Barack Obama's signature domestic achievement.

Trump even called Graham at 10:30 p.m. Monday to emphasize that health care must get done this time around, according to pool reports.

Though it seemed as if the effort to repeal the Affordable Care Act was dead in the water after the so-called "skinny" repeal bill failed this summer with a dramatic late-night vote against it by McCain , there is a renewed push to pass a repeal before the end of the month.

House Speaker Paul Ryan has said he would bring up the bill to the House floor should the Senate pass it, calling it the "best, last chance" the GOP has to overhaul Obamacare.

Pence, who has a day full of meetings at the United Nations General Assembly in New York, flew back to Washington along with Graham for the Republican policy lunch Tuesday, where he will state that the administration is fully behind the latest bill.

Centers for Medicare and Medicaid Services chief Seema Verma and Health and Human Services Secretary Tom Price are expected on the Hill.

Verma, who was crucial in the House vote earlier this year and attempted to play a similar role in the failed July Senate efforts, is back and is considered the point person on getting Murkowski comfortable with the legislation, two aides say. Verma is expected to work on potential proposals to ease the block grant burden on Alaska, though specifics are still up in the air.

According to pool reports, both Trump and House Speaker Paul Ryan phoned for Pence during the flight back to Washington. As Pence took the calls, Graham told reporters he had found an unlikely ally in former White House chief strategist Steve Bannon, who he said has helped to push the latest health care effort over the finish line.

"I have got Alan Greenspan, Jeb Bush and Steve Bannon" behind this bill, Graham told reporters. "If anyone can do better I'd like to meet them.

Bannon, the former White House chief strategist who was fired last month, has also worked to galvanize support for the bill. A source familiar said he has lobbied senators and pushed for favorable coverage on Breitbart News, the website he runs.

Where the swing votes are

The Graham-Cassidy bill would eliminate federal funding for Medicaid expansion and for Obamacare subsidies that lower premiums, deductibles and co-pays in 2020. States would receive a lump sum of money annually through 2026.

Collins, the Maine Republican who voted against a repeal bill earlier this year, said that she has a "number of concerns," including the fact the bill would restructure Medicaid in a fundamental way without considering the ramifications. For Maine, it would be $1 billion less funding over a decade, she said. People would preexisting conditions would also be hurt.

"It seems to have many of the same flaws of the bill we rejected previously and in fact, it has some additional flaws because there's some language that leads me to believe that people worth preexisting conditions would not be protected in some states," Collins said.

The Congressional Budget Office also hasn't reviewed the new bill and likely won't before the end of the month.

"It's difficult not having a CBO analysis to rely on," she noted.

Sen. Rand Paul, R-Kentucky, called the bill "another big government boondoggle." Paul, who eventually backed the GOP bill this summer, bashed the plan on Fox News Tuesday

"This isn't a repeal. This is keeping Obamacare and redistributing the proceeds," Paul said. "So, this is not a repeal bill, this is sort of, 'Hey, we'll take Obamacare, replace it with Obamacare, but we're going to let the states have a little more power in how we spend it.'"

Sen. Ted Cruz, who has not made up his mind on Graham-Cassidy, but is open to the bill, told reporters "steady progress" is being made.

"I think we are making steady progress. I would say the discussion today at lunch was encouraging," he said. "We are headed in the right direction."