WASHINGTON—The search for Donald Trump’s next Supreme Court nominee is beginning where it left off last year, with some candidates the president strongly considered before settling on now-Justice Neil Gorsuch, people close to the process say.

Mr. Trump’s ultimate choice to succeed Justice Anthony Kennedy, who this week announced his retirement after 30 years on the court, is unlikely to be a complete surprise: as a candidate he pledged to select Supreme Court nominees from lists prepared by leaders of conservative organizations,...

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