Senate Majority Leader Mitch McConnell (R., Ky.), was seen after a closed-door meeting with Republican lawmakers to advance the GOP tax overhaul, on Capitol Hill in Washington, on Dec. 1, 2017.

WASHINGTON—Senate Republicans on Friday were confident that they had locked in enough votes to pass a sweeping tax overhaul later in the day. Wavering GOP senators offered their support and a holdout who worried it would add to the deficit all but conceded defeat.

Sen. Jeff Flake (R., Ariz.) Friday afternoon became the latest holdout Republican to move his name into the ‘yes’ column. The Arizona Republican announced his support hours after Sens. Ron Johnson (R., Wis.) and Steve Daines (R. Mont.) made similar declarations. Sen. Susan Collins (R., Maine) still hadn’t committed to the legislation, but said she was making progress in reaching her goals. That likely leaves Sen. Bob Corker (R., Tenn.) as the sole Republican expected to vote against the tax bill.

“We have the votes,” Senate Majority Leader Mitch McConnell (R., Ky.) told reporters.

It could put Republicans one step closer to the dramatic reshaping of the tax code they have been talking about for years and deliver a legislative victory for President Donald Trump. The bill would cut tax rates on corporations, remove or limit longstanding tax breaks and subject fewer wealthy families to the estate tax.

Republicans control 52 seats in the Senate, compared with 48 for Democrats. No Democrats are expected to support the bill, leaving Vice President Mike Pence to break a tie if needed. Passage would clear the way for negotiations with House lawmakers to iron out differences with a competing version that already passed the other chamber.

Much could still change in a fluid negotiating environment. The emerging deal marks the latest in a series of sharp reversals that have already taken place in the past few days.

The Senate bill will keep the alternative minimum tax for individuals and the lesser-known one for corporations, a GOP aide said.

That would remove one of the big simplifications that Republicans have been touting. They had pledged to repeal the AMT, a parallel tax system originally designed to prevent people from using legal breaks to avoid all taxes.

The change in the GOP would increase the exemption for the individual AMT, meaning it would apply to fewer people. The AMT denies deductions for state taxes, so repealing the state income tax deduction would also shrink the AMT’s reach.

Earlier this week, a trio of Republican deficit hawks were working on a plan to automatically reverse tax cuts if the Republican tax plan failed to pay for itself through stronger economic growth. When that idea ran afoul of parliamentary rules Thursday, Republicans discussed abandoning a key pillar of their plan—permanent reductions in the corporate tax rates from 35% to 20%—to keep the deficit hawks on board.

That switched by Friday morning. Instead the group of hawks split. One member of the group, Sen. James Lankford (R., Okla.), remained firmly in support of the tax package despite his concerns, and then Mr. Flake joined him. That left out Mr. Corker as a potential ‘no’ vote on a bill the Joint Committee on Taxation estimates will add $1 trillion to deficits over the next decade. A permanent 20% corporate tax rate was back in.

“I realize that there’s probably enough votes right now to pass it,” Mr. Corker said Friday. The senator, who has emerged as one of his party’s most steadfast deficit hawks, said he expected the tax cut would probably pass without his support. “I think so,” Mr. Corker said when asked. “I’m certainly not trying to rally people in a different direction.”

Momentum for the tax legislation picked up Friday morning after Republicans also gained support from Mr. Johnson and Mr. Daines, helping the party regroup after Thursday’s stumbles.

Before Friday, Messrs. Johnson and Daines had stopped short of offering their support for the tax bill, holding out for better treatment for “pass-through” entities like partnerships and limited liability companies that pay taxes through the tax returns of individual owners, rather than through corporate returns.

Under an agreement reached with Senate Republican leaders, pass-throughs that qualify for the special treatment will now have a top rate of just under 30%. Their plans to vote for the tax bill were outlined by aides to both men.

Changing the tax rate for qualified pass-through businesses is an expensive endeavor. By Friday, Senate Republicans appeared to have settled on a way to raise new revenues to support that plan. Senators would set rates for one-time repatriation of overseas profits to 7% for illiquid assets, up from the 5% originally proposed, and 14% for cash instead of the 10% originally envisioned, Mr. Daines said. That matches the rates set in a companion House bill.

Susan Collins (R., Maine) remains an uncertain vote. She had yet to commit her support Friday morning, but has said she wants to find a way to back it. On Friday she said Republicans were making “very good progress,” and later elaborated, saying that there was an agreement on her demand to allow for a property-tax deduction of up to $10,000. The original Senate bill had eliminated deductions for all state and local taxes.

If the bill passes the Senate Friday it must then be reconciled with a competing bill that has already passed the House of Representatives. The two measures have some notable differences, including a lower top individual tax rate in the Senate bill, fewer individual tax brackets in the House bill, and differing approaches to estate taxes, taxation of foreign income and treatment of pass-throughs.

Friday marked a big recovery for Republicans eager to score a legislative victory on a signature issue for the party.

On Thursday, the nonpartisan JCT released an analysis stating the Senate plan wouldn’t meet the party’s goal of paying for itself through stronger economic growth. The bill would generate $458 billion in revenue from economic growth and add an extra $51 billion in interest costs over a decade, leaving the net cost of the bill at about $1 trillion over a decade, the JCT analysis found.

That report left GOP leaders scrambling to lasso the 50 votes they need to pass the measure as they tried to come up with a way to placate senators concerned about budget deficits.

Their first attempt to placate the deficit hawks, including Messrs. Corker and Flake, fell short due to procedural challenges. They had wanted to add into the bill a trigger that would set tax increases to kick in if projected growth didn’t materialize. That ran afoul of the Senate parliamentarian, and Thursday evening Senate leaders halted votes on the bill.

By Friday, Mr. Corker indicated that he thought the idea of establishing such a trigger had collapsed.

“I think it probably is,” Mr. Corker said. “When you’ve got 50 votes…which is what I thought all along.”

Other senators being watched were Marco Rubio of Florida and Mike Lee of Utah, who want to make a child tax credit more generous. They have proposed pairing that with a higher corporate tax rate, but are exploring other options.

Mr. Johnson also won agreement for rules to make it easier for pass-through entities to make a transition into corporations, an aide said.

Before the bill passes, Republicans are expected to turn to rapid-fire amendment votes on the Senate floor. Under this process, any senator can get a vote on any amendment.

Republicans have filed amendments to expand charitable deductions, repeal the estate tax and change the tax rules for marijuana. They are constrained by their decision to limit the amount of tax cuts to $1.5 trillion over 10 years and by the changes Mr. Corker and others demanded on Thursday.

Republicans did secure more favorable treatment for big-ticket purchases of equipment and other business property. Under the bill that passed the Senate Finance Committee, for five years, companies could immediately deduct the entire cost of any such property placed in service after Sept. 27. Under an emerging agreement, the more-favorable expensing treatment would phase out over time, shrinking to 80% of the cost of the equipment in 2023 and then declining further to 60% and lower in later years, a GOP aide said.

The core of the fast-moving GOP tax bill was a permanent corporate tax cut combined with tax reductions for individuals and pass-through businesses such as partnerships that expire after 2025. The bill would also repeal the mandate for individuals to have health insurance and allow drilling in the Arctic National Wildlife Refuge.

If the bill passes the Senate, Republicans must reconcile it with a version approved earlier by the House, likely through a conference committee composed of lawmakers from both chambers.

The House and Senate would each then have to vote again. An alternative would be for the House to vote up or down on the Senate bill, but House leaders have insisted that they won’t do that.