Before his ouster, Mr. Lewandowski helped Mr. Trump notch his first primary victory, in New Hampshire, a win that helped vault him to the nomination. Mr. Lewandowski would hope to run on the same outsider energy that Mr. Trump channeled that year.

Just how formidable Mr. Lewandowski would be is a source of disagreement among political professionals, most of whom predict an ugly Republican primary race and general election in New Hampshire.

His opponents would be almost certain to raise questions about his business activities since Mr. Trump took office. In the years since he was fired, Mr. Lewandowski co-wrote two books about Mr. Trump with David Bossie, the head of the conservative group Citizens United. But he has also been an adviser to companies that have interests with the government, and he would be required to file financial disclosure forms that would reveal the extent of those business arrangements.

In May 2017, Mr. Lewandowski left the lobbying firm he helped start, Avenue Strategies, amid increasing questions about whether he was lobbying without having registered. Since then, Mr. Lewandowski has had private clients and for many months gave paid speeches to different groups. Mr. Lewandowski has previously said he is not a lobbyist, and that he has never called government officials on behalf of a client.

Democrats are also looking to highlight allegations of assault made against Mr. Lewandowski. In 2016, during the campaign, he was accused of grabbing a reporter for Breitbart News who was trying to ask Mr. Trump questions at one of his properties in Florida. Mr. Lewandowski was charged with simple battery by the police. The charges were later dropped. The following year, a pop singer accused Mr. Lewandowski of slapping her at a party in Washington.

And Gov. Chris Sununu of New Hampshire, a Republican who is up for re-election next year, has told the White House he has concerns about the impact that Mr. Lewandowski’s candidacy could have on others on the ballot, according to two people briefed on the discussions.

“Corey Lewandowski is a shadow lobbyist with a nefarious client list that includes foreign interests, which is why New Hampshire Republicans and even Governor Chris Sununu have been keeping their distance,” said Josh Marcus-Blank, a spokesman for the New Hampshire Democratic Party. “Lewandowski, with his record of violence, will make an already nasty Republican primary even worse.”

The G.O.P. primary already has several declared candidates, including the retired Brig. Gen. Donald C. Bolduc and Bill O’Brien, the former state house speaker. After Mr. Bossie released a private poll he had commissioned from a pollster working with Mr. Trump’s campaign that showed Mr. Lewandowski leading in a theoretical Republican primary, aides to Mr. Bolduc released their own internal survey showing Mr. Bolduc ahead.