A co-founder of the Podesta Group, Tony Podesta, told his team to operate on the understanding that Mr. Yanukovych “is the client,” while an employee at the other firm, Mercury Public Affairs, called the claim that the nonprofit was independent from Mr. Yanukovych “nonsense,” comparing it to “Alice in Wonderland.”

Nonetheless, the Podesta Group and Mercury went along with Mr. Manafort’s ruse, despite the reservations now coming to light. Relying on the written attestation from the European Center, which paid them more than $1 million each, the firms initially registered their engagement with the center under less rigorous congressional lobbying disclosure rules, and not under the foreign lobbying disclosure rules. It was not until last year, when the work came under scrutiny from the Justice Department — about three years after it ended in 2014 — that the firms retroactively registered their work for the European Center under the more rigorous Foreign Agents Registration Act.

Now, that work, and the decision not to disclose it under the Foreign Agents Registration Act, has turned Podesta and Mercury into subjects of interest in a series of linked investigations that have roiled Washington’s lobbying industry by exposing the sometimes surreptitious ways in which foreign interests try to buy influence.

The investigations were prompted by the inquiry of the special counsel, Robert S. Mueller III, into Russian meddling in the 2016 presidential election.

This year, Mr. Mueller’s team referred cases involving possible illegal Ukrainian lobbying by Podesta, Mercury and the law firm Skadden, Arps, Slate, Meagher & Flom — all of which Mr. Manafort had recruited to do work related to Ukraine — to federal prosecutors in New York. And the evidence that Mr. Mueller’s prosecutors publicly unveiled on Friday could help their New York colleagues build the cases against the firms, including the correspondence that questioned the independence of the European Center. The firms were not named in the court filings, but they do not dispute that prosecutors are referring to them.