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 associates, according to the people. The investigation, led by the Manhattan U.S. attorney’s office and the Federal Bureau of Investigation, has already led to campaign-finance charges against the associates, Lev Parnas and Igor Fruman.

A concern of the investigation is whether Mr. Giuliani violated federal lobbying laws by serving as an unregistered agent of a foreign government or hid his work for foreign nationals, said one person familiar with the investigation. Mr. Giuliani has denied ever serving as a lobbyist or agent of a foreign government.

Prosecutors are seeking documents and correspondence with Mr. Giuliani, Giuliani Partners—a security-consulting firm founded by Mr. Giuliani in 2002—or “any related person or entity,” according to one of the subpoenas. The subpoena also seeks documents “related to any actual or potential payments, or agreements” with Mr. Giuliani.

The subpoenas offer the clearest indication yet that federal prosecutors are examining Mr. Giuliani’s consulting work. Giuliani Partners has had several foreign clients, including a city in Ukraine. The subpoenas also sought information on a company co-founded by Mr. Parnas that paid Mr. Giuliani for business and legal advice.