Mr. Giuliani’s company has provided advice and training in policing, terrorism prevention, intelligence and emergency response to governments in South America, the Middle East and Eastern Europe, among other regions, for years. It is currently pursuing work in the Democratic Republic of Congo, as well as a contract with an infrastructure fund focused on Latin America.

Walter M. Shaub Jr., former director of the federal Office of Government Ethics, said the way Mr. Giuliani’s visit to Bahrain was portrayed highlights “the possibility of blurring the line between his personal business activities and official government matters.”

Mr. Shaub said that even without being covered by government ethics rules, Mr. Giuliani “would want to be mindful of serious appearance concerns if he was concerned that his activities might create an impression that he has been authorized to offer access to the president or favorable action by the White House in exchange for business opportunities.”

The confusion also underscores the gap between political norms in the United States versus those in many parts of the Middle East, Africa, Asia and the former Soviet states, where overt efforts to buy favor are more commonly accepted as the way things work.

The perception that such an approach can yield success in Washington has been exacerbated in the Trump era, veteran lobbyists say, because of Mr. Trump’s inclination to value the counsel of a small coterie of friends, relatives and allies over the federal government’s experts.

In Mr. Giuliani’s case, the perception is amplified because of his overlapping roles as a public figure in his own right, a longtime friend of the president’s and the lead lawyer on a relatively small team upon whom Mr. Trump is relying to save his very presidency.

Mr. Giuliani says he has gone to great lengths to make clear to clients and prospective clients that hiring him will not earn any favor from Mr. Trump. And, while he acknowledged that some foreign leaders might try to project otherwise to their domestic audiences by publicizing their photographs or affiliation with him, he said “I can guarantee you, whether they think the picture is valuable, there is nothing I’m going to do to help them other than what they pay me for.”