But the prospects for comprehensive tax reform also remain far from certain given differences in priorities among House and Senate Republicans and an array of business groups prepared to fight provisions that would hurt them.
It is also unclear whether Trump, who has struggled to navigate Capitol Hill, supports the strategy of including the repeal in a broader tax bill, which is likely to include corporate and middle-class tax cuts.
"Republicans are going to have enough problems getting tax reform done," said Jim Manley, a longtime aide to former Senate minority leader Harry Reid, D-Nev. "If they start loading it up with unrelated stuff like this one to score political points, it will just get bogged down and go nowhere."
The measure at issue is named for Lyndon B. Johnson, who introduced it in the Senate in 1954, nine years before he became president. The provision prohibits tax-exempt organizations, including houses of worship, from endorsing or opposing political candidates.
Repeal of the amendment has been sought primarily by conservative Christian leaders, who argue that it is used selectively to keep them from speaking out freely in church.
During the campaign, Trump spoke out in favor of ending the prohibition and he strongly reiterated his support during the National Prayer Breakfast shortly after taking office.
White House spokeswoman Natalie Strom said Trump still supports repeal, but she would not comment on whether he backs the approach being advocated by Rep. Kevin Brady, R-Texas, chairman of the House Ways and Means Committee.
Ralph Reed, chairman of the Faith and Freedom Coalition, which counts repeal of the Johnson amendment as a top priority, said he is not as concerned with how the repeal happens as he is that Trump and other Republican leaders keep their promise.
"That would be fine with us, if it were to become law as part of a tax package," Reed said. "We'd like to have an up-or-down vote, but this might make it easier to pass."
Houses of worship make up just a fraction of the universe of 50-1(c)(3) organizations, named for a portion of the tax code.
Last week, nearly 4,500 such organizations signed onto a letter to congressional leaders, urging them not to weaken or repeal the amendment. It also argued that allowing nonprofit organizations to support candidates would create a loophole in campaign disclosure laws because contributions to many such groups are not made public and are tax deductible.
"Nonpartisanship is a cornerstone principle that has strengthened the public's trust of the charitable community," the letter said.
Fred Wertheimer, president of Democracy 21, one of the groups that signed onto the letter, criticized any attempt by House leaders to attach repeal of the Johnson amendment to broader tax legislation.
"It's simply an effort to ram it through without getting a vote on it," Wertheimer said.
The aim of repealing the Johnson amendment is also not universally embraced by all religious groups. A coalition of 99 organizations, including many Jewish and Baptist groups, sent a separate letter to Congress last week, urging the ban stay in place.