Ed Rollins, a veteran Republican strategist who ran Ronald Reagan’s 1984 re-election campaign and who was a consultant with Teneo until this year, said, “The ability to bring business and political leaders together to meet the C.E.O.s was all part of the selling point.”

Insiders and Email

The office of Mr. Band, a Teneo co-founder and longtime aide to Mr. Clinton, offers a reminder of the firm’s roots. One photograph pictures Mr. Band, Mr. Clinton and President Obama playing golf. Another shows Mr. Band and Mr. Clinton in 2009 after they negotiated the release of journalists from North Korea.

Teneo was formed in June 2011, when Mr. Kelly, the former fund-raiser for Mrs. Clinton, joined with Mr. Band and Paul Keary, who worked with Mr. Kelly at another communications firm. Mr. Keary and Mr. Kelly are members of a rock band at Teneo, Insane Asylum, which performs Rolling Stones covers, former employees say.

Early on, Mr. Clinton was named to Teneo’s advisory board. A search of LinkedIn profiles shows that a number of former aides to the Clintons or employees of the Clinton Foundation have worked there or acted as consultants, including Justin Cooper, an assistant to President Clinton who helped set up the family’s private email server.

One employee who has faced particular scrutiny is Ms. Abedin, who worked as a consultant at Teneo in late 2012 while simultaneously working as Mrs. Clinton’s aide in the State Department. Late last year, a letter sent to Teneo by Senator Charles E. Grassley, Republican of Iowa, chairman of the Senate Judiciary Committee, asked for details of Ms. Abedin’s agreement and whether the arrangement had resulted in undue influence on Mrs. Clinton’s diplomatic work.

Teneo declined to provide the information and said it had provided the necessary details, including Ms. Abedin’s contract, to the State Department Office of the Inspector General, which was conducting its own inquiry into Ms. Abedin’s work status. “Out of respect for the independence and integrity of that review, I respectfully refer your office to them for any other questions you might have,” Mr. Kelly wrote last October.

The State Department Office of the Inspector General declined to comment on the inquiry.

About the same time, Teneo changed its retention policy. In a June 2015 email to staff members that was reviewed by The New York Times, Teneo said that nearly all emails older than one year would be “automatically removed” from its system. The firm cited cybersecurity concerns.