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…

For weeks, Republicans have publicly identified what they see as potential conflicts among Mr. Mueller’s team of more than a dozen investigators. In particular, they have cited thousands of dollars of political donations to Democrats, including former President Barack Obama, made by Andrew Weissmann, a former senior Justice Department official who has expertise in fraud and other financial crimes. News reports have revealed similar donations by other members of Mr. Mueller’s team, which Mr. Trump’s allies have cited as evidence of political bias. Another lawyer Mr. Mueller has hired, Jeannie Rhee, represented the Clinton Foundation.