Newer lobbying upstarts like Ballard Partners have won foreign clients by capitalizing on ties to President Trump. Founded by Brian Ballard, a lobbyist based in Tallahassee, Fla., who was a leading fund-raiser for Mr. Trump’s campaign, transition and inauguration, Ballard Partners has signed contracts with government entities or political parties in Albania, the Dominican Republic, Kosovo and Turkey, all since the inauguration.

Those firms, which have not been accused of impropriety, disclosed the foreign contracts to the Justice Department under the Foreign Agents Registration Act. The law requires anyone who lobbies or does public relations for foreign individuals, companies, governments and political parties to reveal detailed information about their work and payments for it. The act is intended to prevent foreign actors from surreptitiously influencing American public policy.

But for years, many American consultants have skirted the law’s requirements. Mr. Mueller’s inquiry has focused on such undisclosed lobbying among some of its leading practitioners, including Mr. Manafort, who helped pioneer the model for international consulting three decades ago. The investigation has left other firms rushing to ensure their own compliance.

“This is creating concern in the Washington lobbying community about not only complying with FARA, but also a new emphasis on enforcement that perhaps folks didn’t think was happening or would happen,” said Thomas J. Spulak, a lawyer in the Washington office of King & Spalding who specializes in lobbying compliance.

Mr. Mueller has scrutinized Skadden on two fronts — its own work for Mr. Yanukovych’s government, and its role advising to two other powerhouse Washington lobbying firms paid to bolster his government, Mercury Public Affairs and the Podesta Group.

Mr. Mueller accused Mr. Manafort and Mr. Gates of orchestrating a complicated scheme through which Mercury and Podesta were paid about $1 million each through a nonprofit front group to advocate on behalf of Mr. Yanukovych’s government.

As part of his plea agreement, Mr. Gates admitted that he knowingly misled Skadden — though the firm is not named in court filings — in a scheme to avoid FARA registration by indicating incorrectly that the nonprofit was not supervised or controlled by Mr. Yanukovych’s government. Relying on that information, Skadden prepared a legal opinion in 2012 that the lobbying firms used in opting not to register under FARA.