Mr. Gates first heard about Psy-Group’s work during a March 2016 meeting at the Mandarin Oriental hotel along the Washington waterfront with George Birnbaum, a Republican consultant with close ties to current and former Israeli government officials. Mr. Gates had joined the Trump campaign days earlier with Paul Manafort, his longtime business partner, to try to prevent a revolt of Republican delegates from Mr. Trump toward Mr. Cruz, who was the favored candidate among the party’s establishment.

According to Mr. Birnbaum, Mr. Gates expressed interest during that meeting in using social media influence and manipulation as a campaign tool, most immediately to try to sway Republican delegates toward Mr. Trump.

“He was interested in finding the technology to achieve what they were looking for,” Mr. Birnbaum said in an interview. Through a lawyer, Mr. Gates declined to comment. A person familiar with Mr. Gates’s account of the meeting said that Mr. Birnbaum first raised the topic of hiring an outside firm to conduct the social media campaign.

The special counsel’s office indicted Mr. Manafort and Mr. Gates last year on multiple charges of financial fraud and tax evasion. Mr. Gates pleaded guilty to several of the charges this year, and he is cooperating with Mr. Mueller’s investigation.

It is unclear whether the Project Rome proposals describe work that would violate laws regulating foreign participation in American elections. Psy-Group hired Covington & Burling, a Washington-based law firm, to conduct a legal review. Stuart Eizenstat, a former American diplomat and a partner at the firm who participated in the legal review, declined to comment on its conclusions.

Mr. Birnbaum was a protégé of Arthur J. Finkelstein, the legendary Republican political operative, and has spent years as a consultant working on behalf of candidates in foreign elections. In 1996, he helped Mr. Finkelstein engineer Benjamin Netanyahu’s victory over Shimon Peres to become the prime minister of Israel.