A federal tax investigation into Hunter Biden is gaining momentum as prosecutors gather information from several of his associates about the sources of his foreign income, including from Ukraine, and examine President Biden’s son’s relationship with a company that handled some of his finances, according to people familiar with the matter.

In recent weeks, prosecutors from the U.S. attorney’s office in Delaware have sought information and grand-jury testimony about the money Mr. Biden received several years ago from Ukrainian natural-gas company Burisma Holdings Ltd., and how he used that money to pay some obligations, one of those people said.

Last month, prosecutors also extensively questioned at least one other associate of Hunter Biden about Mr. Biden’s drug and alcohol use, spending habits and state of mind in 2018, another person said, suggesting prosecutors are exploring whether such activity would present a defense against a potential criminal tax case.

Prosecutors often seek to get such testimony on the record to secure it before the defense gets a chance to present it more favorably at trial.