Mr Trump also says he has reached out to Mr Khashoggi's fiance, Hatice Cengiz, and invited her to the Oval Office.
"I think that would be hurting us", Trump said.
The Post reported intelligence intercepts of Saudi officials discussing the plan offer evidence that implicate the regime in Khashoggi's disappearance last week after he entered the Saudi Consulate in Istanbul, Turkey.
Kaye said he did not have "any particular plan" about what the global investigation would look like, but that "multiple models" - such as a United Nations -mandated probe - exist.
Senate Foreign Relations Chairman Bob Corker told CNN that the "intel points directly" to Saudi Arabia for the disappearance of Washington Post columnist Jamal Khashoggi and said the Saudis need to "produce" the 59-year-old father of four to dispel concerns. And we will get to the bottom of it.
National security adviser John Bolton and senior adviser Jared Kushner, Trump's son-in-law, talked with Salman on Tuesday, while Secretary of State Mike Pompeo had a follow-up call with the Saudi leader to reiterate the USA demand for information about the case, the White House said.
Erdogan told reporters on a flight back from Hungary that Turkey is anxious about the Khashoggi's disappearance.
Riyadh has dismissed the allegations as "baseless", saying Mr Khashoggi left the consulate alive shortly after he arrived.
Corker said that he had warned the kingdom that its political support in the Senate is "the lowest ever" and that if Saudi officials are responsible for Kashoggi's disappearance, "this would drop it off a cliff".
More than 20 Republican and Democratic senators instructed Trump to order an investigation into Khashoggi's disappearance under legislation that authorizes imposition of sanctions for perpetrators of extrajudicial killings, torture or other gross human rights violations.
U.S. intelligence officials reportedly intercepted communications that the Saudis discussed a plan to lure and capture Khashoggi before his disappearance. A friend of a Saudi journalist who went missing in Istanbul says officials told him to "make your funeral preparations" as the Washington Post contributor "was killed" at the Saudi Consulate in Istanbul.
University of Texas law professor Steve Vladeck weighed in on this revelation, saying it's "remarkable" that the US had this intelligence and seemingly failed to act on it.
According to his fiancée Hatice Cengiz, Khashoggi first arrived at the Saudi Consulate in Istanbul on September 28.
Turkish spokesman Hami Aksoy said authorities are "open to cooperation" and willing to examine the consulate grounds, though it was not clear when a search would take place.
WWE's continued business in Saudi Arabia remains a hot-button topic in professional wrestling, particularly with Crown Jewel (2 November) right around the corner, and PWTorch.com's Wade Keller has this week highlighted a damning new Washington Post article on current USA/Saudi relations.
Reporters Without Borders said in a statement that between 25 and 30 professional and non-professional journalists are now detained in Saudi Arabia. Relations were already strained after Turkey sent troops to the Gulf state of Qatar previous year in a show of support after its Gulf neighbors, including Saudi Arabia, imposed an embargo on Doha. Two appeared to be members of the Saudi royal guard, pictured in a photograph next to Crown Prince Mohammed Bin Salman.
The administration also relies on Saudi support for its Middle East agenda to counter Iranian influence, fight extremism and support an expected peace plan between Israel and the Palestinians.