“It is a simple and elegant way to get good tax compliance,” said Douglas Holtz-Eakin, a Republican economist and president of the right-leaning American Action Forum, a nonprofit tied to a “super PAC” that backs House Republicans.

Some conservatives oppose it for the same reason: In their view, such a tax would be too easy to increase, with the potential costs to Americans hidden behind rising prices.

Groups like Americans for Tax Reform — headed by Grover Norquist, perhaps Washington’s most famous anti-tax crusader — have praised the border tax proposal, saying it would put American businesses “on a level playing field” with foreign competitors. Retailers that import many of their goods are lobbying against the idea, while domestic manufacturers like Boeing and Caterpillar — whose interests figure heavily in Mr. Trump’s economic thinking — are supporting it.

The Koch network and groups like the Club for Growth, which for years have targeted what they call “crony capitalism” in Washington, have opposed the border tax as an unnecessary tax increase and a form of favoritism that would hurt the economy. But Mr. Trump and his team have pledged to target what they see as a more insidious kind of cronyism, including unfettered free trade that some Trump advisers say benefits wealthy elites at the expense of American workers.

The dispute echoes Mr. Trump’s battles with his party last year, when the Club for Growth, a group of wealthy conservatives that backs anti-tax candidates in Republican primary races, financed a multimillion-dollar advertising campaign against him. The Koch network, uncomfortable with Mr. Trump’s proposals on trade and immigration, sat out the presidential election entirely, turning its advertising dollars and activists to down-ballot races.

Both the Club for Growth and the Koch network also played a critical role in killing a proposal backed by Mr. Ryan and Mr. Trump to repeal and replace the Affordable Care Act. In March, as the repeal vote approached, two Koch-aligned groups pledged to spend upward of $1 million on ads defending any Republican who voted against the replacement legislation.