The top tax rate for individuals would drop by a few percentage points, from 39.6 percent to the "mid-30s", according to an official with knowledge of the plan.
USA lawmakers have for years vowed to adopt broad tax reforms, but the efforts have foundered.
During the campaign, Trump called for the rates in the three brackets to be lower than the House plan, at 10, 20 and 25 percent.
Small businesses would see their top tax rate go from 39.6 percent to the proposed corporate tax rate of 15 percent, said Treasury Secretary Steven Mnuchin in a Wednesday morning speech.
And nor will Mr Trump's big infrastructure improvement plan feature.
Mnuchin said the proposed overhaul would amount to "the biggest tax cut" and the "largest tax reform" in USA history.
Small businesses, which include a lot of pass-throughs, worry that Congress will reduce their tax breaks to pay for lower corporate tax rates, but won't lower the top rate for pass through owners and shareholders much, if at all.
The first piece of that policy - "destination-based" - is where the border adjustment comes in.
Trump's plan will not include the controversial tax for now, though it might be revisited later, a person briefed on the rollout told The New York Times.
Under such a provision, companies could no longer deduct the cost of their imported goods, and the sales of their exports would no longer be subject to USA tax.
Without a proposal on the table, the White House has been vague about the president's support for ideas circulating in Congress.
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A border adjustment tax would fundamentally alter how imports and exports are taxed.
Critics of the policy argue it would hurt retailers and consumers because tons of imported products - cars, clothing, appliances - would suddenly become more expensive. And in the case of millions of low- and middle-income families, the breaks would raise their tax burden when combined with Trump's other proposals to eliminate head of household status, repeal personal exemptions and raise the lowest income tax rate to 12% from 10% now.
The prospect of tax cuts without more revenues or budget cuts is already the subject of congressional concern.
This means if taxes for corporations or individuals are cut, they must be offset by some combination of revenue increases elsewhere.
Separately cutting the top tax rate for pass-through businesses, which account for most USA companies, could benefit Trump himself, said Frank Clemente, executive director of Americans for Tax Fairness, a Democratic activist group.
The official scorekeeper for Congress said Tuesday that a big cut in corporate taxes - even if it is temporary - would add to long-term budget deficits.
There's a disparity in the numbers, but the economic theory in both plans is the same: tax cuts will pay for themselves because they spur economic growth.
The U.S. economy, the world's largest, grew at a tepid 1.6 percent pace last year, a figure Trump is hoping to boost to 3 percent a year, which the United States has not reached since 2005. "I think it's pretty clear we're going to have to use a reconciliation vehicle because today's Democratic party is very different from the Democratic party in the 80s", McConnell said.
This is important because the Senate can not pass legislation that would add to long-term budget deficits without a 60-vote majority.
Sen. Orrin Hatch, R-Utah, chairman of the Senate Finance Committee, said he isn't opposed to tax cuts in general, but whacking corporate taxes to 15% could be too aggressive.
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