In the White House's Green Room, President Donald Trump lavished praise on a longtime supporter and friend: billionaire investor Stephen Schwarzman, who was at 1600 Pennsylvania Ave. that day to celebrate the signing of the "phase one" trade deal with China.

In the ornate parlor, Trump dubbed the financier "the great Steve Schwarzman" and Schwarzman responded in kind, thanking the president and calling the Jan. 15 signing of the deal "an important day."

But over the next several months, the two allies could find themselves at odds over an obscure piece of legislation that may prove instrumental to the president establishing his populist bona fides on the campaign trail, and could cost Schwarzman millions of dollars: surprise medical billing.

The term refers to a problem that policymakers say is becoming increasingly more common: patients receiving bills for medical procedures months later and learning that they aren't covered by insurance.

Putting an end to surprise medical billing is one of the White House's top health-care priorities this spring, along with cutting prescription drug prices.

But the policy push could put Trump at odds with his longtime friend and advisor. The White House declined to comment for this article.

Blackstone Group owns a company called TeamHealth, a physician practice firm that specializes in emergency medicine. The company, which was purchased by Blackstone in a deal valued at $6.1 billion in 2016, has been criticized for sending its own "surprise" bills to patients who received emergency care outside of their insurance plan.

TeamHealth and another firm, Envision Healthcare, spent more than $28 million on an ad campaign last year fighting legislation to end surprise medical billing – without disclosing who was paying for the ads, The New York Times reported.

The ad campaign was sponsored by a group called "Doctor Patient Unity," and said big insurance companies have a "scheme to profit from patient's pain" through rate setting. Blackstone Group opposes one of the bills making its way through Congress that would set payments from insurers at a certain level.

At the White House, officials worry that patients are increasingly getting walloped by surprise medical bills. Standing up for voters who are grappling with high medical bills also plays well at the polls.

Blackstone says it supports ending surprise medical billing, and is backing one of several proposals now working its way through the House.

"Blackstone's portfolio company TeamHealth fully supports federal legislation to end surprise medical bills through an approach called independent arbitration, which has proven successful at the state level and has wide, bi-partisan support in Congress," said a spokesman for the firm. "The company also has a long-standing policy against balance billing patients."

But it's not clear that the Trump administration will end up backing the position supported by Schwarzman's firm. Arbitration is just one option being discussed on the Hill, the other is a "benchmark" proposal that is sometimes referred to as "rate setting" and would limit fees.

A senior administration official told CNBC that both Schwartzman's preferred option and the benchmarking idea his firm has lobbied against are under consideration.

"We continue to have discussions with the Hill regarding several options including a market based benchmark and arbitration," the official said, adding that arbitration was not the White House's "originally preferred" policy outcome. "We have concerns about the overuse of arbitration and will be closely monitoring any changes to the legislation," the official said.

Blackstone's critics say the arbitration compromise the firm is backing tilts too far toward doctors and hospitals – and therefore would be the most favorable to Blackstone's company of any of the measures gathering steam on Capitol Hill.

A spokesman for TeamHealth dismissed an alternate compromise provision focused on "rate setting" as overly beneficial to large insurance companies.

"Big insurance companies have spent $74 million lobbying Congress for a rate setting law that would be a huge financial benefit to them while harming patients and their ability to see a doctor," he said. "Rate setting stalled because surprise medical bills can't be fixed with doctor shortages, hospital closures, and loss of access to care."

A senior Democratic aide said the surprise-billing legislation – whatever form it takes – is one of just a small number of items with bipartisan support that could survive an election year and impeachment politics and get signed into law.

"The question is whose ox is getting gored, and in what quantity," said the aide, who didn't want to be identified discussing political strategy. "There is a general interest in getting it done."

All that sets up an unusual political dynamic: Aides for Trump and House Speaker Nancy Pelosi could end up working on the same side of the debate, while Trump splits from his ally Schwartzman.

Still, there's a general wariness on the part of Hill Democrats of dealing with the volatile and unpredictable Trump White House, and a suspicion that the president may switch sides even as the bill is still moving.

"Nobody has any illusions about the nature of this White House," the senior Democratic aide said. "There's not a lot of faith to be put into staff pronouncements of where Trump is."

House aides expect to mark up surprise medical billing legislation in February ahead of a May 22 deadline for the expiration of various so-called health extenders provisions that impact Medicare and Medicaid that were passed in the end-of-the-year spending bill.

As for Schwarzman himself, there's no word on whether he discussed surprise billing with anyone at the White House during his recent visit.