Subpoenas issued to people with ties to President Trump’s personal lawyer, Rudy Giuliani, indicate a broad federal investigation into possible money laundering, obstruction of justice and campaign-finance violations and show that prosecutors are probing Mr. Giuliani’s consulting businesses and other sources of income, according to people familiar with the matter.

In recent weeks, prosecutors have sent subpoenas and other requests to potential witnesses seeking records and information related to Mr. Giuliani and two of his...