WASHINGTON—The Republican-led Senate Intelligence Committee in recent weeks subpoenaed Donald Trump Jr., the president’s eldest son, to testify about answers he gave congressional investigators as part of their Russia probe, according to people with knowledge of the matter, marking the first time a congressional panel has subpoenaed a member of President Trump’s family.

The subpoena—news of which came a day after Senate Majority Leader Mitch McConnell (R., Ky.) declared the Russia investigation “case closed”—is likely to further...