Sens. Marco Rubio and Jeff Flake, members of the Foreign Relations Committee, said Congress is prepared to move quickly and firmly if Trump fails to adequately respond to the October 2 disappearance of Khashoggi, a Washington Post contributor.
Trump floated his idea shortly after speaking by phone with Saudi Arabia's King Salman earlier in the day, and he indicated that the king may have hinted at rogue killers. Until now, Riyadh has not allowed Turkish investigators to search the consulate - officially Saudi territory - with reports both sides were at odds over the conditions.
A Turkish diplomatic source had earlier said a joint Turkish-Saudi team would search the building - the last place Khashoggi was seen before he vanished on October 2.
Seasoned observers of Middle East politics, including some at senior levels of the Turkish government, have speculated for days about the likelihood that the royal court would seek to accuse a "rogue" operator within the Saudi security services of killing Khashoggi.
Turkey believes the journalist was murdered with his body removed, while Saudi Arabia has dismissed such allegations.
Turkey and Saudi Arabia work together on some common issues and share business interests, but also have a good deal of political bad blood.
But a Turkish diplomatic source said it was expected that the search, a joint operation with Saudi authorities, would "take place towards the evening" Monday.
The controversy has troubled Saudi's traditional Western allies - who are key arms suppliers to the kingdom - and also undermined efforts by Mohammed bin Salman to present himself as a modernising ruler.
Turkish sources allege he was killed by a 15-strong team of Saudi agents but Riyadh insists that he left the consulate unharmed. "This is very important to get to the bottom of it", Trump said.
But CNN cited two sources as saying the Saudis are preparing a report that will acknowledge Khashoggi's death resulted from an interrogation that went wrong during an intended abduction.
Saudi Arabia has begun an internal investigation into the disappearance of a prominent journalist at its Istanbul consulate and could hold people accountable if the evidence warrants it, according to a Saudi official.
Britain, Germany and France have jointly called for a "credible investigation, ' Egypt has backed Saudi Arabia and warned of "false claims" against its ally and Australia's foreign minister has said she is 'deeply concerned". " In it, author S.V. Date quotes Courtney Radsch of the Committee to Protect Journalists and former Central Intelligence Agency analyst Ned Price to claim Trump's rhetoric likely led to Khashoggi's disappearance".
Saudi Arabia has strongly denied any involvement in the reporter's disappearance and claimed Khashoggi left the consulate, but provided no evidence to back up their claims.