Sen. Joe Manchin III (D-W.Va.) told the White House last week that he would support some version of a tax targeting billionaire wealth as part of President Biden’s Build Back Better economic agenda, according to three people familiar with his private offer.

Despite its endorsement from the most conservative Democrat, the tax on billionaires still faces long odds to approval as part of the final legislation, as it has been greeted skeptically by other Democratic lawmakers in both the House and Senate. It was also left out of the House Democrats’ version of Build Back Better after Manchin publicly criticized it in October.

Manchin’s offer to the White House — details of which The Washington Post reported earlier this week — included a list of spending and revenue proposals that he supports. Manchin listed the tax on billionaire wealth as an option toward the bottom of his list, the people said. It is unclear whether Manchin’s plan included a revenue estimate. The people spoke on the condition of anonymity to discuss the private offer.

Manchin’s inclusion of the measure reflects the potential common ground that remains between the president and the senator in his party most uneasy about the White House’s sprawling economic agenda, which could cost as much as $2 trillion over 10 years and remake significant parts of the American economy. Alan J. Auerbach, an economist at the University of California at Berkeley, has been advising Manchin personally about tax policy, two people familiar with the matter said. Auerbach declined to comment.

But other considerable obstacles remain.

For months, Manchin has been largely supportive of the White House’s tax and revenue plans, focusing his criticisms on spending measures he says are poorly designed and that would trigger more inflation and explode the national debt. Those concerns have yet to be resolved. But Manchin is not the only Democrat with policy issues that remain unaddressed.

Sen. Kyrsten Sinema (D-Ariz.), long the biggest hurdle to Democrats’ tax aspirations, has again in recent days raised concerns about some of the revenue measures the party is pursuing.

In particular, Sinema has questioned whether owners of “pass-through” entities — companies structured so the owner “passes through” income onto their personal income tax returns — should be exempted from a new “surtax” intended to fall on the very rich, two people familiar with the matter said. These people spoke on the condition of anonymity to reflect private conversations.

The White House was forced to dramatically revamp its tax proposals after Sinema previously ruled out increasing the corporate tax rate, which Biden initially sought to raise from 21 percent to 28 percent. To make up for the lost revenue, the White House and Sinema agreed to a number of measures, including the new “surtax” that would impose a 5 percent levy on top of income earned above $10 million in a given year and an additional 3 percent levy on income above $25 million per year. Roughly 22,000 people, or 0.02 percent of U.S. households, would be affected by the measure.

The surtax as drafted would apply to both ordinary income and capital gains income. Exempting pass-through entities from that measure could deprive Democrats of billions of dollars they are counting on to ensure their plan does not add to the national debt. It is unclear whether Sinema will make the request conditional for advancing the broader economic legislation.

Spokespeople for Manchin and Sinema declined to comment. The White House also declined to comment.

While the specifics of what Manchin would support remain unclear, Senate Finance Committee Chairman Ron Wyden (D-Ore.) has unveiled a tax aimed at the accrued wealth of America’s approximately 700 billionaires. The measure is aimed at addressing the massive gains of the wealthiest Americans with a federal tax and probably would be unprecedented in how few people it affected. Some Democrats, including House Speaker Nancy Pelosi (D-Calif.), have expressed concerns that the measure was hastily drafted and not thoroughly vetted. Congress’s nonpartisan scorekeeper has estimated it could raise as much as $550 billion over 10 years, or pay for more than one-quarter of the Democrats’ spending bill.

Sinema has been supportive of the tax measure on billionaires after extensive conversations with Sen. Elizabeth Warren (D-Mass.). The New York Times reported in October that Manchin criticized the billionaire tax by saying he did not “like the connotation that we’re targeting different people,” adding that many of these people “create a lot of jobs and invest a lot of money and give a lot to philanthropic pursuits.”