Should he run, Mr. Lewandowski would be a high-octane test case for an emerging campaign dynamic: former Trump associates running for office.

Sarah Huckabee Sanders is reportedly mulling a run for governor of Arkansas in 2022. Speculation about a possible presidential bid continues to surround Nikki Haley, Mr. Trump’s former U.N. ambassador, who’s been taking small steps to distance herself from the president. And some Republicans are trying to push Secretary of State Mike Pompeo into the Senate race in Kansas…

Trump supporters in New Hampshire argue that a loyalist could nationalize the profile of a second-tier Senate race and give their base the boost of enthusiasm it needs in a state the president lost by just 2,736 votes in 2016.

But some Republicans in the state are fiercely opposed to Mr. Lewandowski’s candidacy, leaving them in rare open revolt to the White House. Many are concerned with the baggage he would bring to the race: He was accused of grabbing a female reporter when he worked for the campaign, and he has drawn scrutiny over his “advisory” business, which let him represent clients while remaining close to the White House.