In a note affixed to the top of the column, Post Global Opinions editor Karen Attiah said she received the essay from Khashoggi's translator and assistant a day after he was reported missing.
Over the past few days, Mr Trump has raised the possibility of "rogue killers" being behind the journalist's disappearance. "I'm going to allow the process to move forward", he said.
On Wednesday and into Thursday, investigators spent nearly nine hours searching the Saudi consul's residence, then moving on to the consulate itself about 200m (650ft) away, according to Reuters news agency.
In the column, Khashoggi, a Saudi citizen who went into self-imposed exile in the USA over the rise of the crown prince, also discussed the practice of Middle Eastern governments blocking internet access to control tightly the information their citizens can see.
Turkish government sources have said police believe the journalist was killed by a special team of 15 Saudi officials sent to Istanbul especially for the task.
Several of the Saudi suspects identified by Turkish officials reportedly have direct links to the 33-year-old crown prince, with whom the Trump administration has built close ties.
Ms Attiah said the column was a flawless example of Khashoggi's commitment and passion for freedom in the Arab world, which was "a freedom he apparently gave his life for".
Confirming that the tape said to provide evidence of the killing had been requested, Mr Trump added: "I'm not sure yet that it exists, probably does, possibly does".
Turkish authorities say Khashoggi was killed.
"You know we need Saudi Arabia in terms of our fight against all of the terrorism, everything that's happening in Iran and other places".
"They also have assured me they will conduct a complete, thorough investigation of all of the facts surrounding Mr. Khashoggi and do so in a timely fashion and that the report itself will be transparent", Pompeo told reporters today on the White House lawn, moments before Treasure Secretary Steve Mnuchin announced he would not participate in next week's MBS-hosted "Davos in the Desert" conference.
The paper said a Saudi team immediately grabbed the journalist after he entered the consulate, cutting off his fingers and later decapitating him.
Turkish crime-scene investigators had entered the residence of the Saudi consul general on Wednesday to search for evidence in the disappearance of Saudi writer Jamal Khashoggi, just as Turkey's daily Yeni Safak published a gruesome recounting of his alleged slaying.
In his brief comments to reporters, Pompeo also stressed the "longstanding strategic relationship" between the USA and Saudi Arabia.
The New York Times had reported that Mutreb had travelled extensively with the Crown Prince, perhaps as a bodyguard. He also confirmed that all members of his family support the official Saudi investigations, which only "will lead us to positive results and uncover the truth".
Meanwhile, International Trade Minister Liam Fox has been told to pull out of a business conference in Saudi Arabia next week as finance ministers from France and the Netherlands and prominent banking executives have scrapped their plans to attend.