WASHINGTON—The highest-earning Americans will get a lower tax rate and corporations will pay slightly more than in previous plans under a deal House and Senate Republicans reached on the party’s competing tax-overhaul bills.

Full details of what is likely to be a $1.4 trillion tax cut over a decade will be released this week. If the House and Senate both pass the measure in votes that could come next week, President Donald Trump could sign it into law before Christmas.

“We’ve got a pretty good deal,” Senate Finance Committee Chairman Orrin Hatch (R., Utah) told reporters in the Capitol on Wednesday.

Republicans hashed out the agreement without input from Democrats in advance of a formal meeting of the bipartisan House-Senate conference committee on Wednesday, and they announced the deal just before heading to a lunch at the White House with Mr. Trump. Democrats complained about the closed process, but to no avail.

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The agreement would set the top individual tax rate at 37%, two people familiar with the deal said. That is lower than today’s 39.6% and lower than the top rate in each of the bills that passed the House and Senate.

Republicans said they were considering that change to address concerns from high-earning residents in high-tax states like California, New York and New Jersey who would take a hit because they would lose the ability to fully deduct state and local taxes as part of the overhaul. Rather than fully repeal the deduction as planned, the compromise would cap it at $10,000, a GOP aide said.

The corporate rate would be 21%, the people said. That is higher than the 20% rate Republicans included in the House and Senate tax bills. It would take effect in 2018. The Senate bill had delayed that rate cut—from today’s 35%—until 2019.

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Mr. Trump, who had said he wanted a 15% rate, then drew a red line at 20% before expressing openness to going as high as 22%, on Wednesday said he “would be thrilled” with a 21% corporate rate.

“We want to give you the American people a giant tax cut for Christmas. And when I say giant, I mean giant,” Mr. Trump said in a speech at the White House on Wednesday. “Our current tax code is burdensome, complex and profoundly unfair.”

The agreement is also expected to eliminate the corporate alternative-minimum tax, the people said. Keeping that, as the Senate bill did, would have undercut the value of many popular business-tax breaks, including a research-and-development tax credit. The bill would retain the individual alternative minimum tax with exemptions for incomes up to $500,000 for individuals and $1 million for married couples, much higher than current law.

Republicans are considering a 20% income deduction for so-called pass-through businesses such as partnerships and S corporations, which pay taxes through individual returns. That is lower than the 23% deduction in the Senate bill, but when combined with the lower top tax rate on ordinary income, it equates to a nearly identical 29.6% top rate on that income for pass-throughs. The bill also would include some version of House language letting pass-through firms qualify for a tax break based on their capital investment, a GOP aide said.

The final agreement is expected to steer clear of some of the more controversial changes in the House plan, including taxes on graduate-student tuition waivers, the repeal of deductions for student-loan interest and medical expenses and the end of tax-free private activity bonds used for projects such as hospitals and affordable housing.

It was unclear how Republicans made these changes and stayed within the $1.5 trillion tax-cut limit they set for themselves.

House and Senate negotiators have been working this week to reconcile the tax-cut bills that have passed each chamber and had been narrowing their differences. Republicans are now working on writing the legislative text of the tax bill, getting official revenue estimates and making final decisions. Votes on the final proposal could start as early as Monday as Republicans try to wrap up their work before they leave Washington for the year. The Senate is expected to vote first.

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The core of the tax bill is likely to remain the same as the versions passed by the House and Senate: a deep corporate rate cut, a restructuring of the international tax system, cuts in individual income-tax rates, and big benefits to high-income business owners and heirs to large estates. Middle-income households will get tax cuts that are set to expire, and some households, particularly upper-middle-class residents of high-tax states, would likely pay more than they do now.

The tax plan would encourage modestly faster economic growth but not enough to cover its own costs, according to the nonpartisan Joint Committee on Taxation. The JCT estimated that the House and Senate bills would each add $1 trillion to budget deficits over the next decade, even after accounting for the benefits of faster economic growth likely to come from lower taxes.

Business groups voiced cautious optimism about key provisions affecting corporate taxes, praising the decision to cut the corporate tax rate starting next year as well as the elimination of the corporate alternative minimum tax. But they expressed reservations about other provisions that could still surface in the final bill.

“Unquestionably, these guys have to be jumping for joy,” said Andrew Schmidt, a North Carolina State University accounting professor who specializes in taxes. “I can’t think of any business-oriented group that has not been pushing for this for the past 15 years.”

The plan has been faring poorly in public-opinion polls, but Republicans are determined to press ahead, seeing its enactment as a political imperative heading into 2018 elections and as their preferred way to spur the economy.

House Minority Leader Nancy Pelosi (D., Calif.) said the plan is a loser for the GOP, whether it succeeds or fails, noting the pressure the GOP’s wealthy donors have placed on getting a tax cut.

“If they don’t pass the bill, they lose with their paymasters,” she said. “If they do pass the bill, they lose with the American people.”

The vote on the bill will be tightest in the Senate, which passed its tax-cut measure 51-49 this month. Several Republicans have since expressed concerns about the emerging agreement.

Sen. Marco Rubio of Florida wants to see larger increases in child tax credits beyond those that are in either the Senate or House bill.

Late Wednesday, Sen. Mike Lee (R., Utah) told constituents that the degree to which a $2,000 child-tax credit would be available to the working poor under a new tax bill was still being negotiated. “A significant portion—we don’t yet know for sure how much of that—will be refundable up to the total amount of taxes paid, including the payroll taxes,” Mr. Lee said to participants in a telephone town-hall meeting.

Sen. Susan Collins of Maine has said she opposes lowering the top individual rate.

“I’m going to wait and look at the entire conference report once it comes out,” Ms. Collins said on Wednesday. “There are a lot of parts still moving.”

Sen. Bob Corker of Tennessee, who was the lone Republican to vote against the Senate version, said the bill is likely to include some unspecified changes he has sought. His major concern about the fiscal cost of the plan appears unlikely to be addressed, though Mr. Corker hasn’t said yet how he will vote on the final bill.

Republicans face even more of a time crunch to pass their bill quickly after Democrat Doug Jones won an upset victory in Tuesday night’s Alabama Senate race. The result will shrink the GOP Senate majority to 51-49, but Mr. Jones isn’t expected to be seated until January. Democrats on Wednesday called for the tax-bill votes to be delayed until them, but Republicans are sticking to their timeline.

“It would be wrong for Senate Republicans to jam through this tax bill without giving the newly elected senator from Alabama the opportunity to cast his vote,” said Senate Minority Leader Chuck Schumer (D., N.Y.).

—Kristina Peterson, Rebecca Ballhaus, Eli Stokols and Theo Francis contributed to this article.

Write to Richard Rubin at richard.rubin@wsj.com and Siobhan Hughes at siobhan.hughes@wsj.com