George Nader, a Lebanese-American businessman, has hovered on the fringes of international diplomacy for three decades. He was a back-channel negotiator with Syria during the Clinton administration, reinvented himself as an adviser to the de facto ruler of the United Arab Emirates, and last year was a frequent visitor to President Trump’s White House.

Mr. Nader is now a focus of the investigation by Robert S. Mueller III, the special counsel. In recent weeks, Mr. Mueller’s investigators have questioned Mr. Nader and have pressed witnesses for information about any possible attempts by the Emiratis to buy political influence by directing money to support Mr. Trump during the presidential campaign, according to people with knowledge of the discussions.

The investigators have also asked about Mr. Nader’s role in White House policymaking, those people said, suggesting that the special counsel investigation has broadened beyond Russian election meddling to include Emirati influence on the Trump administration. The focus on Mr. Nader could also prompt an examination of how money from multiple countries has flowed through and influenced Washington during the Trump era.