Days after President Biden was sworn in last year, one of his top congressional allies went to the White House with the name of a judge he believed should be appointed to the Supreme Court.

The ally, Representative James E. Clyburn, Democrat of South Carolina, told Vice President Kamala Harris and the White House counsel, Dana Remus, that whenever an opening emerged on the court, Mr. Biden should nominate a little-known federal judge in his home state: J. Michelle Childs.

Mr. Clyburn, who helped Mr. Biden revive his candidacy with a crucial primary win in South Carolina nearly two years ago, made a case that Judge Childs would not only satisfy Mr. Biden’s campaign pledge to appoint a Black woman to the court, but that the judge was particularly appealing because she came from a blue-collar background — another underrepresented group among federal judges.

“One of the things we have to be very, very careful of as Democrats is being painted with that elitist brush,” Mr. Clyburn told The New York Times last year for an article that revealed how he was pushing her for the court.

He added: “When people talk to diversity they are always looking at race and ethnicity — I look beyond that to diversity of experience.”

As Mr. Biden weighs whom to appoint to fill the seat of Justice Stephen G. Breyer, Judge Childs is seen as a potential contender — albeit an outside one — alongside two other possibilities: Judge Ketanji Brown Jackson of the U.S. Court of Appeals for the District of Columbia Circuit and Justice Leondra R. Kruger of the California Supreme Court.

Judge Jackson and Justice Kruger attended Ivy League law schools, unlike Judge Childs, who attended the University of South Carolina.

The president has already moved to give Judge Childs a promotion: He said in December that he would nominate her to the federal appeals court for the District of Columbia Circuit, a frequent staging ground for potential Supreme Court justices. The Senate Judiciary Committee has scheduled a hearing for her confirmation to the D.C. Circuit for next Wednesday, a process that could draw more attention to her and provide an opportunity for her to impress the White House.

Mr. Clyburn, whose endorsement of Mr. Biden at a low point in his run for the Democratic Party’s nomination helped him win South Carolina and propel him to the presidency, is expected to push the White House in the coming days to nominate Judge Childs.

“Not just for our party, but for the judiciary, it’s important to have somebody who has lived experiences,” Mr. Clyburn told The Times last year.

Judge Childs’s mother moved her to South Carolina from Michigan after her father died when she was a child. Her mother worked for telephone companies. Judge Childs attended the University of South Florida as an undergraduate on a scholarship. She then began the climb from the bottom of the legal world to the federal judiciary. In 1992, she became the first female law partner at a prominent firm in South Carolina. She later served as a state judge before President Barack Obama nominated her in 2010 to be a district court judge.

In one of her more high-profile cases, Judge Childs struck down a South Carolina rule during the 2020 election that would have required a witness to sign absentee ballots, a move intended to make it more onerous to vote, particularly during the coronavirus pandemic.