On Sunday, The New York Times reported that Mr. Trump had expressed renewed interest in nominating Judge Hardiman, who was the runner-up to Judge Neil M. Gorsuch as the president’s first Supreme Court nominee in January 2017.
Mr. Trump was inspired by Judge Hardiman’s biography, according to people close to the process, and has considered the recommendation of his sister, Judge Maryanne Trump Barry, who served with Judge Hardiman on the United States Court of Appeals for the Third Circuit.
Judge Hardiman was also, along with Judge Kethledge, one of two finalists Senator Mitch McConnell of Kentucky, the majority leader, recently recommended to Mr. Trump and the White House counsel, Donald F. McGahn II, arguing that they posed the least risk of a confirmation failure.
Mr. McConnell saw drawbacks in the others. Judge Kavanaugh once argued that President Bill Clinton could be impeached for lying to his staff and misleading the public, a broad definition of obstruction of justice that would be damaging to Mr. Trump if applied in the special counsel’s Russia investigation. And Judge Kavanaugh has been viewed with suspicion by conservatives wary of his connection to President George W. Bush, for whom he served as staff secretary.