Here is a look at some early front-runners for the job, and the full list of 25 potential nominees, as released by the White House in November.

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The Front-Runners

Thomas M. Hardiman, United States Court of Appeals for the Third Circuit

President Trump considered Judge Hardiman last year for the seat that was ultimately filled by Justice Neil M. Gorsuch. Judge Hardiman was first appointed to the federal bench by President George W. Bush in 2003 and was elevated to the appeals court four years later. He has built a reputation as a reliable conservative on the court, where he has served alongside Mr. Trump’s sister, Judge Maryanne Trump Barry, who is said to have recommended Mr. Hardiman for the Supreme Court vacancy last year. One opinion that could resonate with Mr. Trump: Judge Hardiman signed on to a decision declaring that asylum seekers could not ask a Federal District Court to prevent or postpone their deportation while challenging their removal orders.

William H. Pryor Jr., United States Court of Appeals for the 11th Circuit

Judge Pryor was viewed as a finalist for last year’s vacant seat on the court. He has called Roe v. Wade, the 1973 Supreme Court decision that established a woman’s constitutional right to have an abortion, as “the worst abomination of constitutional law in our history.” He is close with Attorney General Jeff Sessions and, like Mr. Sessions, is an outspoken conservative who has strongly opposed gay rights.

Amul R. Thapar, United States Court of Appeals for the Sixth Circuit

Judge Thapar was confirmed to the appeals court last year and previously served as a judge on the Federal District Court in eastern Kentucky. The son of Indian-American immigrants, the White House said he was the first federal court judge of South Asian descent. Judge Thapar was among those considered by Mr. Trump for last year’s Supreme Court vacancy. “I’m my own judge and I hope my track record speaks to that,” he told senators last year.