Novartis has said that its former chief executive, Joe Jimenez, entered into the agreement with Mr. Cohen as part of an effort to gain insight into the approach the new administration would take on topics of interest to Novartis, particularly health care. The company said that, after an initial meeting with Mr. Cohen last March, it concluded that he did not have the expertise they had hoped for and decided not to go forward with the arrangement. But it said it learned the contract could only be canceled for cause and allowed it to expire in February.

Since revelations about the relationship last Tuesday, Novartis has sought to distance its new chief executive, Vas Narasimhan, from the controversy, saying that the decision to hire Mr. Cohen had been made by Mr. Jimenez and that Mr. Narasimhan had played no role. Mr. Jimenez retired from the company in January.

In a letter to employees last week, Mr. Narasimhan called the deal a “mistake” that led the company to be criticized “by a world that expects more from us.”

Mr. Jimenez has not responded to requests for comment.

Novartis has said that Robert S. Mueller III, the special counsel investigating Russian interference in the 2016 presidential election, contacted them last November to inquire about the connection to Mr. Cohen, and that the company cooperated with the investigation and considers its role closed. Since last week, several Democratic senators, including Elizabeth Warren of Massachusetts, Richard Blumenthal of Connecticut, Ron Wyden of Oregon and Patty Murray of Washington, sent Novartis letters demanding more information about the deal.

The telecommunications giant AT&T has also acknowledged paying Mr. Cohen $600,000 for a similar arrangement. AT&T has called the deal a “big mistake” and last week said its top Washington lobbyist would be leaving.