The work can carry reputational and legal risks, since clients often come with toxic baggage and the United States Treasury Department restricts transactions with entities under sanctions. As a result, it commands some of the biggest fees of any sector in the influence industry. And some of the biggest payments have been going to lobbyists, lawyers and consultants with connections to Mr. Trump or his administration.

“People overseas often want to hear that you know so-and-so, and can make a call to solve their problem,” said Erich Ferrari, a leading Washington sanctions lawyer who said he has tried to disabuse prospective clients of such notions.

It is a perception that matches up with the pay-to-play mind-set that defines politics in many parts of Africa, Asia, the Middle East and the former Soviet states. As politicians and executives from those regions have increasingly been targeted by sanctions, they have sought to apply that approach — backed by huge sums of cash — to navigating Washington, lobbyists and former government officials say.

This has been encouraged, they say, by the willingness projected by Mr. Trump and his team to make deals around sanctions and tariffs exemptions. Previous administrations had worked to wall off politics from those processes, which are supposed to be overseen primarily by career officials and governed by strict legal analyses.