Rudy Giuliani, the former New York City Mayor and Donald Trump loyalist, has spent the past decade in the private sector traveling the world and making millions by giving speeches and consulting on security issues.

He has spent the last two weeks engaged in an unusually public fight to land the secretary-of-state slot in the next administration. Mr. Trump, the Republican president-elect, is also considering nominating Mitt Romney, the GOP’s 2012 presidential nominee and former investment banker. But Mr. Giuliani and his allies...