WASHINGTON — President Biden’s pledge to name a Black woman to fill a coming Supreme Court vacancy has thrust Republicans into a tricky political calculation, forcing them to confront how aggressive to be in opposing the nominee and how to do so without appearing to be racist and sexist.

While Supreme Court battles have become scorched-earth affairs in recent years, Republicans are weighing whether to wage all-out war or take a more tempered approach against Mr. Biden’s pick, particularly given that whomever the president chooses to succeed Justice Stephen G. Breyer will not change the conservative ideological tilt of the court.

Many of them recognize that a divisive fight could provide more fodder for Democrats to try to deepen the wedge between their party and African Americans before this year’s midterm elections. And while some take issue with Mr. Biden’s pre-emptive promise to name the first Black woman to the court, arguing that the choice should be based on merit rather than race or gender, Republicans enter the coming showdown fully aware that the groundbreaking nature of the president’s pick could make challenging the nominee far more fraught.

“The idea that race and gender should be the No. 1 and No. 2 criteria is not as it should be,” said Senator Susan Collins, Republican of Maine, who is regarded as a potential swing vote in favor of Mr. Biden’s pick. “On the other hand, there are many qualified Black women for this post and given that Democrats, regrettably, have had some success in trying to paint Republicans as anti-Black, it may make it more difficult to reject a Black jurist.”

Others say the historic nomination of the first Black female justice without the philosophical balance of the court in play could provide the opportunity for a reset after a series of confirmations enveloped by brutal partisanship, even if most Republicans ultimately oppose the choice.

“I think there is a lot of value in lowering the temperature,” said Senator Kevin Cramer, Republican of North Dakota.

Even as most Republicans appear inclined to prevent the review of the nominee from veering into a highly polarized and partisan clash, doing so will not be easy. At least two Republican senators, Roger Wicker of Mississippi and Ted Cruz of Texas — white men, like the vast majority of the Senate — have drawn criticism in recent days for suggesting that Mr. Biden’s promise is offensive and akin to affirmative action.

Mr. Wicker said the nominee would be benefiting from a “quota” of the sort the Supreme Court itself has reviewed, while Mr. Cruz said Mr. Biden’s pledge was “actually an insult to Black women.”

Senator Elizabeth Warren, Democrat of Massachusetts, said such comments showed that the Republican Party had moved on from “racial dog whistles and gone directly to racial sirens.”

Senator Richard J. Durbin, Democrat of Illinois and the chairman of the Judiciary Committee, said on Monday that as the Biden administration has sought to expand the diversity of the federal bench over the past year, he has found that the nominations of “assertive women of color” have drawn more fire from Republicans than he would have anticipated.

Still, Mr. Durbin said he held out hope that the eventual nominee would be able to attract bipartisan support, and said he had been actively reaching out to Republicans to assure them that they would have access to the person chosen.

“I think there are certain Republican senators who may — may — consider voting for a Biden nominee,” he said.

While the court process is in its very early stages, the first few days have been notably subdued compared with the charged atmosphere that marked recent confirmation clashes from the very start.

On Tuesday, Mr. Durbin and Senator Charles E. Grassley of Iowa, the senior Republican on the Judiciary Committee, headed to the White House for what appeared to be an amicable meeting with Mr. Biden to discuss the pending nomination.

The president, a former chairman of the panel, said he welcomed Republican input. “I’m serious when I say I want the advice of the Senate as well as the consent,” he said.

Later, aides to Senator Mitch McConnell of Kentucky, the Republican leader, released a routine statement saying that he had spoken to Mr. Biden by phone and shared his desire for a nominee who demonstrated “a commitment to originalism and textualism.”

It was a sharp contrast to 2016, when Mr. McConnell went to the White House to inform President Barack Obama personally that he had no intention of considering Mr. Obama’s nominee to fill the seat left open by Justice Antonin Scalia’s death, a meeting the Senate leader later called one of his proudest moments.

Both parties immediately went to battle stations, a dynamic that persisted in ugly Supreme Court confirmation fights in 2018 and 2020.

This time, the leaders of both parties have been more reserved.

“You can anticipate the Senate, the Republican minority, treating the nominee with respect and going through the process in a serious, thoughtful way,” Mr. McConnell told reporters on Tuesday.

In his own remarks, Senator Chuck Schumer, Democrat of New York and the majority leader, alluded to Republican attacks on Mr. Biden’s pledge to make his first nominee a Black woman.

“Every single member of this chamber, regardless of party, should embrace the president’s commitment to make sure our courts — and especially the Supreme Court — better reflect our country’s diversity,” Mr. Schumer said. “And nominating a Black woman as a justice is a long overdue step towards achieving that goal.”

Senate Republicans are particularly sensitive to suggestions of racism at the moment. Democrats have spent the past several weeks accusing them of ignoring efforts to suppress minority voting in Republican-led states and equating their opposition to federal legislation to counter such efforts with “Jim Crow 2.0.”

Mr. Biden directly compared opponents of the Democratic voting rights package that fell to a Republican filibuster last month to notorious Southern racists such as Bull Connor. That comparison drew an angry and aggrieved response from Republicans such as Mr. McConnell and Senator John Thune of South Dakota, the party’s No. 2.

However, Senate Republicans, with their almost entirely white and male membership, are also aware that they can be susceptible to such attacks. They added two Republican women to the Judiciary Committee after they were forced to bring in a female prosecutor in 2018 to avoid the spectacle of the all-white male panel challenging Christine Blasey Ford over her sexual misconduct allegations against Justice Brett M. Kavanaugh.

Brian Fallon, the executive director of the progressive group Demand Justice who has closely followed the review of Biden judicial nominees, said the backlash to the comments from Mr. Cruz and Mr. Wicker illustrated the dangers ahead for Republicans if they took too hard a line.

“Those two just emphasize that Republicans are not very skillful articulating opposition to a Biden nominee who is the first Black woman without it coming across as being race-based,” he said. “They are very clumsy at this.”

With conservative voters traditionally motivated by Supreme Court showdowns, many Republicans will still want to make known their opposition to what will most likely be an ideologically liberal candidate. They say they will not be dissuaded from a deep review of the nominee’s qualifications and philosophy.

“If he nominates somebody who is way out of the mainstream and not a constitutionalist and is objectionable, then who knows?” said Senator Josh Hawley of Missouri, a Republican member of the Judiciary Committee who is considered to have presidential aspirations. “For my part, I think we need to do this — we need to thoroughly vet this person.”

At this point, Republicans say they do not see themselves employing procedural roadblocks such as boycotting Judiciary Committee proceedings to try to derail a nominee, an approach that could be seen as trying to unfairly subvert the process.

They also say they have been more civil to Supreme Court nominees than Democrats have, pointing to the character attacks that they contend marred Justice Kavanaugh’s confirmation process but omitting mention of Mr. McConnell’s nearly yearlong blockade of Mr. Obama’s nominee, Merrick B. Garland.

“Our Democratic friends always seem to make a bigger stink than we do when it comes to nominees on the high court,” said Senator John Cornyn, Republican of Texas and a senior member of the Judiciary Committee. “I would expect a more dignified process.”

Even though they might not end up voting for the nominee, other Republicans said the process could ultimately provide a moment to be relished.

“We fully expect the nominee not to be a conservative,” Mr. Cramer said. “If they are acceptable in every other way and they are a Black woman because the president made a commitment to nominate a Black woman, we should celebrate that.”

Emily Cochrane contributed reporting.