Erik Prince, the private security contractor and the former head of Blackwater, arranged the meeting, which took place on Aug. 3, 2016. The emissary, George Nader, told Donald Trump Jr. that the crown princes who led Saudi Arabia and the United Arab Emirates were eager to help his father win election as president. The social media specialist, Joel Zamel, extolled his company’s ability to give an edge to a political campaign; by that time, the firm had already drawn up a multimillion-dollar proposal for a social media manipulation effort to help elect Mr. Trump.

The company, which employed several Israeli former intelligence officers, specialized in collecting information and shaping opinion through social media.

It is unclear whether such a proposal was executed, and the details of who commissioned it remain in dispute. But Donald Trump Jr. responded approvingly, according to a person with knowledge of the meeting, and after those initial offers of help, Mr. Nader was quickly embraced as a close ally by Trump campaign advisers — meeting frequently with Jared Kushner, Mr. Trump’s son-in-law, and Michael T. Flynn, who became the president’s first national security adviser. At the time, Mr. Nader was also promoting a secret plan to use private contractors to destabilize Iran, the regional nemesis of Saudi Arabia and the Emirates.