Federal prosecutors have stepped up a criminal investigation into possible efforts by longtime Republican fundraiser Elliott Broidy to profit from his connections to the Trump administration, according to people familiar with the matter.

Investigators are examining whether Mr. Broidy, a top fundraiser for Mr. Trump in 2016 who was later tapped as deputy finance chairman of the Republican National Committee, was paid by his intelligence-contracting firm’s foreign and prospective clients to give them special access to attend...