Mr. Gowdy’s chief interest, according to people briefed on the inquiry, is a series of memos that Mr. Blumenthal — who was not an employee of the State Department — wrote to Mrs. Clinton about events unfolding in Libya before and after the death of Col. Muammar el-Qaddafi. According to emails obtained by The New York Times, Mrs. Clinton, who was secretary of state at the time, took Mr. Blumenthal’s advice seriously, forwarding his memos to senior diplomatic officials in Libya and Washington and at times asking them to respond. Mrs. Clinton continued to pass around his memos even after other senior diplomats concluded that Mr. Blumenthal’s assessments were often unreliable.