The fact that Mr. Gates was allowed to plead guilty to just two relatively lower-level charges indicated to legal experts that he must have something of value for Mr. Mueller. The presumption in Mr. Trump’s circle is that Mr. Gates may not have any incriminating information about the president but could be a dangerous witness against Mr. Manafort, who in turn could threaten Mr. Trump.

Mr. Manafort participated in a meeting in June 2016 along with Donald Trump Jr., the candidate’s son, and Jared Kushner, his son-in-law, with a Russian lawyer on the promise of receiving incriminating information about Hillary Clinton on behalf of Russia’s government.

Mr. Manafort also reportedly offered during the campaign to give “private briefings” to Oleg V. Deripaska, a Russian oligarch allied with President Vladimir V. Putin who claimed Mr. Manafort owed him $19 million. Prosecutors are interested in learning how a Republican convention platform plank on Russia’s intervention in Ukraine was watered down.

On the other hand, Mr. Trump’s defenders said Mr. Gates’s credibility as a witness may be tainted by the fact that one of the charges he pleaded guilty to was lying to the F.B.I. even as he was negotiating his plea deal. Ty Cobb, the White House special counsel, has said that Mr. Manafort has no damaging information against Mr. Trump. Mr. Manafort insisted again on Friday that he was innocent and would fight the “untrue piled up charges.”

Mr. Trump’s argument that none of this has anything to do with him resonates with many of his supporters, who have echoed his repeated insistence that there was no collusion with Russia. But being surrounded by people who are prosecuted has damaged other presidents even when they were not directly implicated.

Jimmy Carter endured significant political damage when his confidant and budget director, Bert Lance, was accused of banking irregularities. Even putting aside the Iran-contra scandal and the Monica S. Lewinsky affair, Ronald Reagan and Bill Clinton were besieged by investigations of their aides unrelated to them. In Reagan’s era, it became known as the “sleaze factor.” Mr. Clinton’s team drew the scrutiny of six independent counsels other than Kenneth W. Starr.

In the current case, the targets so far have included not just a “coffee boy,” as Mr. Papadopoulos was described by an adviser to Mr. Trump, but the president’s top two campaign officials and national security adviser. While Mr. Trump has dismissed the relevance of allegations against Mr. Manafort because they involved business dealings before the campaign, the latest indictment claims that he was scheming to defraud banks while serving as Mr. Trump’s chairman.