Riyadh said on Saturday that Khashoggi died during a "brawl" inside the consulate after a visit on October 2 to get documents.
Three other suspects are linked to Prince Mohammed's security detail and a fifth is a high-level forensic doctor, the New York Times reported.
Trump said Monday he is sending his top diplomat to Saudi Arabia after speaking to Saudi King Salman, who denied knowledge of the fate of missing journalist Jamal Khashoggi.
The warning from the world's top oil exporter came after a turbulent day on the Saudi stock exchange, which plunged as much as 7 per cent at one point Sunday.
The report by the Sabah newspaper, which is close to Turkish President Recep Tayyip Erdogan, published images of the men apparently taken at passport control.
Khashoggi, a former Saudi government adviser, had been living in self-imposed exile in the United States since last year fearing possible arrest.
Jamal Khashoggi, a contributor to the Washington Post, went to the Saudi consulate on Tuesday to receive an official document for his marriage and has not appeared since then.