WASHINGTON—George Conway, the husband of senior White House adviser Kellyanne Conway, is set to be nominated to run the Justice Department’s civil division, according to people familiar with the matter, a job that would put him at the forefront defending the controversial immigration executive order and other lawsuits against the Trump administration.

Mr. Conway, a partner at Wall Street law firm Wachtell, Lipton, Rosen & Katz, had also been in the running for other jobs at the Justice Department.

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