WASHINGTON — Engaging in lawyerly small talk, Senator Hawley earlier this week was telling the Supreme Court nominee, Ketanji Brown Jackson, how he met his wife while clerking at the high court.

Ms. Jackson already knew the story, he discovered. She even “filled in some of the details for me.”

“So I thought — she’s very well prepared.”

Ms. Jackson was prepared, as well, for the Republican senator’s questions about Guantanamo Bay detainees she represented 15 years ago as a public defender and, after that, in private practice. Mr. Hawley said after the meeting that he is still concerned about that part of her record but found her forthcoming and engaging, with a “very high degree” of legal acumen.

“I think her hearings will be very substantive,” he said.

Ms. Jackson, who sits on the federal appeals court and would replace Justice Stephen Breyer, who is retiring, is unlikely to need support from Mr. Hawley or any other Republican to be confirmed, and may not win over any of them. Yet as she makes the rounds at the Capitol, traversing from one Senate office to the next before her confirmation hearings later this month, Ms. Jackson is networking with zeal, restoring a collegial tone to a confirmation process that had grown increasingly embittered during the Trump era.

“I want to make this a bipartisan vote,” the Senate Judiciary Committee chairman, Dick Durbin, said after Justice Breyer announced his retirement. “I think it is not only good for the Supreme Court, it’s good for the Senate.”

Democrats and the White House are hoping that Ms. Jackson’s enviable resume, empathetic style, and historic potential as the first Black female justice will win at least a few crossover votes. And because her confirmation to replace the liberal-leaning Justice Breyer wouldn’t shift the ideological balance of the court, Republicans aren’t expending much political energy opposing her.

Mr. Durbin and President Biden have been reaching out to some GOP senators personally, promising to answer any questions and give them extended time with the nominee.

The most gettable Republican vote is that of Senator Collins, who has already received three calls from Mr. Biden and met Ms. Jackson for more than 90 minutes on Tuesday. Ms. Collins, one of only three Republicans to vote for Ms. Jackson when she was confirmed to the circuit court last year, called the meeting “very productive.” She signaled that the nominee is likely to have her vote.

“She takes a very thorough, careful approach in applying the law to the facts of the case, and that is what I want to see in a judge,” Ms. Collins said.

Ms. Jackson was prepared for the small talk in that meeting, as well, telling Ms. Collins in the first few minutes that she got engaged to her husband in Maine.

“She passed that test,” Ms. Collins joked to reporters in her office as the two women smiled together for the cameras.

Even if other Republicans don’t vote for Ms. Jackson, it’s clear that she has impressed many of them as she has navigated the awkward ritual of the meet and greet. Senator Cornyn praised her experience as a public defender and said she was “charming.” Senator Tillis noted how prepared she was, a move he said was “wise.” Senator Sasse shook her hand and congratulated her while the two smiled for cameras under a large buffalo head on his office wall.

The effusiveness from some Republicans is a sharp departure from the last several Supreme Court nominations.

In 2016, after the death of Justice Antonin Scalia, majority Republicans denied a vote on President Obama’s nominee, Merrick Garland, who is now the attorney general, and most Republicans refused even to meet him. Democrats’ frustration over their jilted nominee was ever-present when President Trump nominated Neil Gorsuch to fill the post the next year.

In 2018, Mr. Trump nominated Brett Kavanaugh to replace Justice Anthony Kennedy; his confirmation came after an explosive, combative hearing in which Christine Blasey Ford accused him of sexually assaulting her in high school, which Justice Kavanaugh denied.

And Democrats had few words of praise for Justice Amy Coney Barrett, who was nominated by  Mr. Trump to replace a liberal icon, Justice Ruth Bader Ginsburg, after her death just weeks before the 2020 presidential election.

Republican reaction to Ms. Jackson hasn’t all been positive. Before Mr. Biden named her as the nominee, several Republicans, including Mr. Hawley, criticized the president’s pledge to nominate a Black woman to the post. Mr. Hawley called the pledge “hard woke left.” Senator Cruz said it was “offensive” to have that criteria. Senator Wicker likened it to affirmative action.

The good vibes could dissipate in the hearings, as well, if race moves to the forefront or if some Republicans make more personal arguments against her. The Senate Republican leader, Mitch McConnell, has so far signaled to his ranks to avoid that kind of questioning and keep the focus on issues they see as more damaging to Democrats, like inflation.

Asked about Mr. Biden’s pledge to nominate a Black woman, Mr. McConnell replied: “Honestly, I did not think that was inappropriate.”

Still, Republicans are sure to question Ms. Jackson aggressively during her confirmation hearings, which begin March 21, and criticize her for any rulings that they see as too far to the left. Messrs. Hawley, Cornyn and Tillis — all members of the Senate Judiciary Committee — said they have questions about whether Ms. Jackson is guided by any specific judicial philosophy.

Democrats who have met with the nominee appear thrilled with Mr. Biden’s choice, gushing over Ms. Jackson’s eight years as a federal judge, her time as a public defender and her ability to connect with others — a quality they say could help her bring the Supreme Court closer together, as well.

Senator Hirono, a member of the Judiciary panel, said that when she met with Ms. Jackson she asked the judge what stood out to her most about Justice Breyer, whom she had clerked for many years ago. She said Ms. Jackson replied that it was the justice’s ability to reach out to other members of the court.

“Even if she can’t convince other people to her way of approaching a case, I think that that willingness to talk, and understanding another perspective, is a very important aspect of the kind of person she is,” Ms. Hirono said.

Another Democrat on the committee, Senator Blumenthal, said after his meeting that Ms. Jackson has a “really engaging personality” as well as a superior intellect.

“What really struck me most was her personal depth and warmth, and her intuitive interest in how real people are affected by her decisions beyond the abstract legalisms,” Mr. Blumenthal said.

By MARY CLARE JALONICK