Veteran Republican operatives are increasingly agonizing over the image of a party whose contenders have been beset by scandals, factual gaffes and a fickle electorate that seems unimpressed by the choices they have been given.

Now, Mr. Trump promises once again to inject his personality in the Republican race as he hosts a debate days before the Jan. 3 Iowa caucuses. But leading Republican strategists and campaign officials on Monday condemned a Trump-moderated debate as a spectacle that would do more harm to the party than good.

The billionaire grabbed headlines this year by seizing on the “birther issue,” questioning whether President Obama was an American citizen, and Mr. Trump continues to dangle the idea of an independent bid for the White House next year.