Subpoenas issued to people with ties to President Trump’s personal lawyer, Rudy Giuliani, and his associates indicate a broad federal investigation into possible money laundering, obstruction of justice and campaign-finance violations, and suggest that prosecutors are looking closely at the work of Mr. Giuliani himself, 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.

The subpoenas offer the clearest indication yet that federal prosecutors are examining Mr. Giuliani’s consulting work. Among the entities named in the subpoenas are Giuliani Partners, a security-consulting firm founded by Mr. Giuliani in 2002 that had multiple 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.