WASHINGTON — The policy debate over raising the federal minimum wage to $15 an hour is the latest fault line between Democrats, who largely support the idea, and Republicans, who generally oppose such a sharp increase as bad for business.

But it is also revealing new fissures in the Republican Party, which is straining to appeal to its corporate backers, some of whom believe that more than doubling the minimum wage would cut deeply into their profits, and the working-class wing, which fueled President Donald J. Trump’s rise and would stand to gain from a pay increase.

After decades of either calling for the abolishment of a federal minimum wage or arguing that it should not be raised, Republicans are beginning to bow to the realities facing the party’s populist base with proposals that acknowledge the wage floor must rise. President Biden is likely to try to capitalize on that shift as he tries to deliver on his promise to raise the minimum wage, even if it does not make it into the $1.9 trillion aid package because of a ruling Thursday evening by the Senate parliamentarian.

For years, Republicans have embraced the economic arguments that were laid out in a letter this month to Congress by Americans for Tax Reform, the Club for Growth and other conservative groups that promote free enterprise. They point to studies that assert mandated wage increases would lead to job losses, small-business closures and higher prices for consumers. And they make the case that the economic trade-offs are not worth it, saying that more jobs would be lost than the number of people pulled from poverty and that those in states with a lower cost of living — often conservative-leaning states — would bear the brunt of the fallout.

In 2016, as Republicans moved further to the right, moderate candidates such as Jeb Bush, a former Florida governor, and Senator Marco Rubio of Florida, argued forcefully that the federal minimum wage did not need to be raised above $7.25, which is where it still stands today. Mr. Bush said the matter of wages should be left to the private sector, while Mr. Rubio warned about the risk of making workers more costly than machines.

But Republicans have at times grappled with the challenging politics of a position that so clearly sides with business interests. In the 2012 presidential campaign, Mitt Romney, the Republican nominee, said that he believed that the federal minimum wage should rise in step with inflation, as measured by the national Consumer Price Index.

And after arguing early on in his 2016 campaign that wages were already too high, Mr. Trump later said he could support a $10 minimum wage.

That is the number that Mr. Romney, now a Republican senator from Utah, and Senator Tom Cotton, Republican of Arkansas, introduced in a plan that would gradually raise the minimum wage to $10 over four years and then index it to inflation every two years.

On Friday, Senator Josh Hawley, Republican of Missouri, went a step further by matching the proposal that Democrats have made for a $15 minimum wage. His plan comes with a big caveat, however, and would apply only to businesses with annual revenue of more than $1 billion.

“Megacorporations can afford to pay their workers $15 an hour, and it’s long past time they do so, but this should not come at the expense of small businesses already struggling to make it,” Mr. Hawley said.

The proposal drew a sharp rebuke from David McIntosh, the president of the Club for Growth, who suggested that Mr. Hawley was adopting bad policies in a bid to appeal to Mr. Trump’s voters. He said that his organization would not support Republicans who promoted minimum wage increases and said that they should be pushing for payroll tax cuts to give workers more take-home pay.

“This is another example of his ambition driving him to these populist positions that completely violate any principles he has about free markets,” Mr. McIntosh said in an interview.

While the talking points surrounding the minimum wage have remained largely the same over the years, the politics are shifting partly because the federal wage floor has stagnated for so long — and a growing economic literature has suggested that the costs of higher wage floors may not be as significant as analysts once worried they might be.

After rising gradually over the decades, the minimum has held steady at $7.25 an hour since 2009. Prices have gradually increased since then, so the hourly pay rate goes a shorter distance toward paying the bills these days: Today’s $7.25 is equivalent to $5.97 in 2009 buying power, adjusted by consumer price inflation.

Given how low it is set, a relatively small share of American workers actually make minimum wage. About 1.1 million — 1.5 percent of hourly paid workers and about 0.8 percent of all workers — earned at or below the $7.25 floor in 2020.

States with the highest share of hourly paid workers earning at or below the federal minimum are often Southern — like South Carolina and Louisiana — and skew conservative. About seven in 10 states that have an above-average share of workers earning at or below the minimum wage voted Republican in the 2020 presidential election.

While only a slice of the work force earns at or below the minimum, lifting the federal base wage to $15 would bolster pay more broadly. The $15 minimum wage would lift pay for some 17 million workers who earn less than $15 and could increase pay for another 10 million who earn just slightly more, based on a recent Congressional Budget Office analysis.

Still, raising wages for as many as 27 million Americans is likely to come at some cost. The budget office, drawing on results from 11 studies and adjustments from a broader literature, estimated that perhaps 1.4 million fewer people would have jobs in 2025 given a $15 minimum wage.

Some economists who lean toward the left have questioned the budget office’s conclusion.

In research that summarized 55 different academic studies of episodes where a minimum wage was introduced or raised — 36 in the United States, 11 in other developed countries — Arindrajit Dube at the University of Massachusetts Amherst found that even looking at very narrow slices of workers who were directly affected, a 10 percent increase in minimum wage might lead to a 2 percent loss in employment. Looking at the effects for low-wage workers more broadly, the cost to jobs was “minute.”

More recent work from Mr. Dube has found next to no employment impact from state and local minimum wage increases.

Yet many Republicans have seized on the budget office’s job loss figure.

In a column titled “How Many Jobs Will the ‘Stimulus’ Kill?” Stephen Moore, an adviser and ally of Mr. Trump’s, and the conservative economist Casey B. Mulligan suggest that the $15 federal minimum wage will cost a million jobs or more. Mr. Moore said in an email that they were relying on the Congressional Budget Office’s estimate.

Still, a variety of economic officials emphasize that the cost to jobs of a higher minimum wage are not as large as once believed, and that the federal minimum wage has not kept up with inflation.

“Higher minimum wages clearly do help the workers who are affected,” John C. Williams, the president of the Federal Reserve Bank of New York, said during a virtual speech on Thursday. “There are some job losses,” but recent evidence suggests that it is not as many as once expected.

There is precedent for raising the minimum wage toward $15, because as the federal base pay requirement has stagnated, states and localities have been increasing their own pay floors. Twenty states and 32 cities and counties raised their minimum wages just at the start of 2021, based on an analysis by the National Employment Law Project, and in 27 of those places, the pay floor has now reached or exceeded $15 an hour.

The drive toward $15 started in 2012 with protests by fast-food workers and was initially treated as something of a fringe idea, but it has gained momentum even in states that are heavily Republican. Florida — which Mr. Trump won in November 2020 — voted for a ballot measure mandating a $15 minimum wage by 2026.

Like in many of those local cases, Democrats are proposing a gradual increase that would phase in over time. Janet L. Yellen, the Biden administration’s Treasury secretary and former Fed chair, suggested in response to lawmaker questions after her confirmation hearing that the long runway could help mitigate any costs.

“It matters how it’s implemented, and the president’s minimum wage will be phased in over time, giving small businesses plenty of time to adapt,” Ms. Yellen wrote.