WASHINGTON — When President Biden nominated Judge Ketanji Brown Jackson to a prestigious appeals court last year, Senator Lisa Murkowski of Alaska, a centrist known for her willingness to break with her party, was one of only three Republicans to vote to confirm her.

Now Ms. Murkowski, who is in a challenging re-election race in the state she has represented for two decades, faces a difficult political predicament as she weighs whether to support Judge Jackson’s confirmation to the Supreme Court, joining Democrats in backing the first Black woman to serve there.

The two other Republicans who supported Judge Jackson for her current post have come down on opposite sides of the question. Senator Susan Collins of Maine said on Wednesday that she would vote to confirm Judge Jackson, calling her qualified and experienced. Senator Lindsey Graham of South Carolina said on Thursday that he would oppose the judge, calling her a liberal judicial activist.

That leaves Ms. Murkowski, who says she is still undecided, and who is among a small but dwindling group of Republicans whom the White House regards as prime targets to support Judge Jackson.

“I know that others have already made their decisions; that’s good for them,” Ms. Murkowski told reporters on Wednesday, hours after Ms. Collins announced her position. The Alaskan indicated that she was not close to a decision ahead of a vote that Democrats are planning for late next week, and that she still intended to “get more into my process.”

Ms. Murkowski’s process is known to be unpredictable, and this year it involves some tricky political calculations.

The three-term senator has the distinction of being the only Senate Republican who voted to convict former President Donald J. Trump in his second impeachment trial who is also facing voters this year. Back home in Alaska, she is confronting a feisty challenger on her right who has been endorsed by Mr. Trump. A “no” vote on Judge Jackson could shore up her standing with conservatives who may have been alienated when she broke with the former president.

But new election rules in Alaska have scrambled the political calculus. For the first time, candidates will compete in an open primary regardless of party, and the four top vote-getters will advance to the general election, where voters will rank them to determine a winner. The system gives candidates an incentive to appeal to the broadest possible constituency in both parties, rather than their own party’s narrow set of core supporters.

For Ms. Murkowski, a vote to confirm Judge Jackson could potentially help her cobble together a coalition of centrist Republicans, independents and Democrats to make up for those on the right who may have abandoned her because of her frequent defections from the party line, and from Mr. Trump.

That has been a winning formula for her in past races, and Ms. Murkowski’s allies argue that it is her likeliest path to re-election.

“There’s no right to shore up,” said Jim Lottsfeldt, a lobbyist and political consultant based in Alaska who is close with Ms. Murkowski. “The people who love Trump will not forgive her for the impeachment vote; it’s a waste of time to chase them.”

There is no public polling yet in the race, and neither candidate is running television ads. Ms. Murkowski’s advisers insist that politics rarely, if ever, play into her votes on judicial nominees. They note that her 2018 vote against Justice Brett M. Kavanaugh, Mr. Trump’s nominee, demonstrated that she is willing to pay a political price to vote her conscience. (Mr. Trump said at the time that she would “never recover” from the vote.)

She tends to keep her own counsel, and her staff often does not know how she plans to vote. Two years after her will-she-or-won’t-she vote on Justice Kavanaugh, Ms. Murkowski voted to confirm Justice Amy Coney Barrett, another of Mr. Trump’s Supreme Court nominees, despite vocally objecting to her party’s rushed process to push through the nomination on the eve of the 2020 election.

“She looks at the record, she looks at the person, she looks at the qualification and talks to them with an open mind,” said Scott Kendall, who previously served as her campaign counselor.

But her precarious political situation has only increased the pressure on Ms. Murkowski when it comes to Supreme Court battles past.

As she looked toward a difficult re-election race in 2010, Ms. Murkowski voted to oppose the confirmation of two of President Barack Obama’s Supreme Court nominees. She registered a surprise “no” on Justice Sonia Sotomayor in 2009, citing the nominee’s decisions in past cases involving the Second Amendment and property rights. A year later, she opposed Justice Elena Kagan, whom she called “evasive.”

She would go on to lose her primary anyway to Joe Miller, a Tea Party candidate, but then mounted a successful write-in campaign from the political center and became the first write-in candidate in more than 50 years to win a Senate election.

This year, Ms. Murkowski has plenty of ground to make up with conservatives in her state, which twice voted overwhelmingly for Mr. Trump. She was censured in 2021 by the Alaska Republican Party for her vote to convict Mr. Trump during his second impeachment trial.

She is being challenged by Kelly Tshibaka, a Trump acolyte who has promoted false claims of election fraud and written articles in support of gay conversion therapy. Mr. Trump, whose midterm political strategy is driven almost completely by a vengeful effort to unseat Republicans who broke with him, has said he plans to campaign in the state for Ms. Tshibaka, who also hired his former campaign managers.

After the sole Democrat in the race dropped out last week, Ms. Murkowski has no challenger on her left.

That means she would pay little political price for voting against Judge Jackson’s confirmation.

“Her biggest challenge would come from the right, or from Trump,” said Senator Kevin Cramer, Republican of North Dakota. “If she voted ‘no,’ it would be hard to blame her.”

The silver lining for Ms. Murkowski is that 60 percent of voters are not registered Republicans or registered Democrats.

Despite the state’s deeply embedded independent streak and the changes to the election rules that make the race less partisan, Ms. Tshibaka has framed the confirmation vote as yet another example of Ms. Murkowski catering to Democrats.

“Alaskans are tired of the same old guessing game, ‘Which way will Murkowski vote?’” Ms. Tshibaka said in a statement. “She’s always torn between doing what’s right for Alaska or catering to her Washington, D.C., elitist friends.”

If elected, Ms. Tshibaka said, she would never support “the leftist nominees and the D.C. insiders.”

Mr. Lottsfeldt said he had watched the senator deliberate on Supreme Court nominations and that she viewed them, in particular, as “out of the realm of ordinary politics.”

“Then you overlay Jan. 6, the lack of civility and the whole Trump experience, and I think those feelings become elevated for her,” he said. Ms. Collins’s decision to vote in favor of Judge Jackson’s confirmation, he said, “makes it so much more safe for Lisa to do what she wants to do. I know Lisa was surprised when they split on Kavanaugh.”

Still, Ms. Murkowski’s conservative supporters in Washington would prefer to have one fewer issue for her to field attacks on. Many remain frustrated by her vote against Justice Kavanaugh, as well as her vote to confirm Deb Haaland, the Biden administration’s interior secretary, arguing that those actions have made her re-election bid that much more difficult.

Yet her record reflects the singular attributes of her unusual state.

Ms. Murkowski has often recounted to friends back home a story about a private meeting of Republican lawmakers where senators were reviewing the political makeup of each state. When Alaska flashed on the screen, showing more than 60 percent of voters not registered as Republicans or Democrats, Senator Joni Ernst, Republican of Iowa, gasped.

“Oh!” she exclaimed. “That’s why you vote the way you do.”

The reality of that map appears to have stuck with Ms. Ernst.

“It is very complicated, but I know Lisa will do what is right for her constituents,” she said in an interview on Wednesday. “I trust her judgment, and she knows her state best.”