WASHINGTON — Robert S. Mueller III, the special counsel, has referred three investigations into possible illicit foreign lobbying by Washington insiders to federal prosecutors in New York who are already handling the case against President Trump’s former lawyer, according to multiple people familiar with the cases.

The cases cut across party lines, focusing on both powerful Democratic and Republican players in Washington, including one whom Mr. Trump has repeatedly targeted — the Democratic superlobbyist Tony Podesta. The cases are unlikely to provoke an outburst from Mr. Trump similar to the one he unleashed in April after prosecutors raided the home and office of Michael D. Cohen, then the president’s lawyer. But these cases do represent a challenge to Washington’s elite, many of whom have earned rich paydays lobbying for foreign interests.

They also tie into the special counsel investigation of Mr. Trump: All three cases are linked to Paul Manafort, the president’s former campaign chairman, whose trial on financial fraud charges began Tuesday in Alexandria, Va.

Mr. Manafort earned tens of millions of dollars lobbying for Ukrainian politicians with ties to Russia, and he drew other Washington lobbyists and lawyers into the work. Now, all three cases being taken on by the United States attorney’s office in Manhattan are examining whether those lobbyists failed to register as foreign agents, as is required by law, and how they were paid, said people familiar with the investigations.