The president of U.S.A. Gymnastics discussed the possibility of a top security job at the United States Olympic Committee with an F.B.I. agent investigating the national team doctor on allegations of sexual abuse.

Steve Penny, the federation president, who stepped down in March 2017, had also worried about the organization’s image and sought to cultivate a close relationship with federal investigators, going so far as to ask for their recommendations on the wording of public statements about the investigation, according to emails reviewed by The New York Times. In one email to an F.B.I. employee, he wrote, “We need some cover.”

Yet Mr. Penny went further in his contacts with Jay Abbott, the special agent in charge of the F.B.I.’s field office in Indianapolis. Because the gymnastics federation is based in Indianapolis, that field office had begun the inquiry into the doctor, Dr. Lawrence G. Nassar. Dr. Nassar, who abused scores of gymnasts during his tenure, is now in prison.

Mr. Penny’s lawyer, Edith R. Matthai, in response to questions about whether Mr. Abbott had been offered a job while the agency was investigating Dr. Nassar, confirmed that the U.S.O.C. position had been discussed but insisted there was no conflict of interest.