A bid by Mr. Lewandowski would have had support from key parts of Mr. Trump’s political apparatus in a rare race where Republicans would have a chance to pick up a Senate seat in 2020. It would also have tested the appetite for Trumpism in a purple state with a Republican governor where Hillary Clinton very narrowly won in 2016 and Ms. Shaheen won her Senate race two years earlier by less than three percentage points.

At least three Republicans are already in the contest: William O’Brien, a former speaker of the New Hampshire House of Representatives; Donald Bolduc, a retired Army brigadier general; and Corky Messner, a businessman and veteran.

In August, Mr. Trump all but endorsed Mr. Lewandowski in an interview with a New Hampshire radio host. “I will say this — if he ran, he would be a great senator. If he ran and won, he’d be a great senator,” Mr. Trump said in the interview.

Mr. Lewandowski helped Mr. Trump notch his first primary victory in New Hampshire in February 2016, which began Mr. Trump’s march to the Republican presidential nomination. Mr. Trump’s children fired Mr. Lewandowski as the campaign manager less than five months later, but the president would retain a fondness for him and would speak with him often.

If he had run in New Hampshire, Mr. Lewandowski’s business activities would have been the subject of intense scrutiny. While Mr. Trump has been in office, Mr. Lewandowski has been an adviser to companies that have interests with the government.