Federal prosecutors in Manhattan have launched a criminal probe into sex-abuse allegations against Harvey Weinstein, said people familiar with the matter, escalating law-enforcement scrutiny of the former Hollywood producer.

One issue in the investigation is whether Mr. Weinstein lured or induced any women to travel across state lines for the purpose of committing a sex crime, an offense that could allow for federal charges, some people familiar with the investigation said. The U.S. attorney’s office for the Southern District...