WASHINGTON — President Trump’s lawyers and aides are scouring the professional and political backgrounds of investigators hired by the special counsel Robert S. Mueller III, looking for conflicts of interest they could use to discredit the investigation — or even build a case to fire Mr. Mueller or get some members of his team recused, according to three people with knowledge of the research effort.

The search for potential conflicts is wide-ranging. It includes scrutinizing donations to Democratic candidates, investigators’ past clients and Mr. Mueller’s relationship with James B. Comey, whose firing as F.B.I. director is part of the special counsel’s investigation.

The effort to investigate the investigators is another sign of a looming showdown between Mr. Trump and Mr. Mueller, who has assembled a team of high-powered prosecutors and agents to examine whether any of Mr. Trump’s advisers aided Russia’s campaign to disrupt last year’s presidential election.

Some of the investigators have vast experience prosecuting financial malfeasance, and the prospect that Mr. Mueller’s inquiry could evolve into an expansive examination of Mr. Trump’s financial history has stoked fears among the president’s aides. Both Mr. Trump and his aides have said publicly they are watching closely to ensure Mr. Mueller’s investigation remains narrowly focused on last year’s election.