Trump said Treasury Secretary Steve Mnuchin's trip to attend a Saudi investment conference is still on but could be canceled by Friday depending on what the investigation finds.
"There was no attempt to interrogate him. They had come to kill him", the source said.
Although Turkish officials have claimed that Khashoggi was murdered, no official explanation has been offered about his whereabouts.
Sky special correspondent Alex Crawford says she was told by a "very well placed" source that a conversation between Saudi consul general Mohammed al-Otaibi and Mr Khashoggi can be heard on the recording, after which there was "a beating".
Turkish officials believe Mr Khashoggi was murdered by Saudi agents but Riyadh has always strongly denied this.
The US chief diplomat was greeted at Riyadh airport by Saudi Foreign Minister Adel Al-Jubeir and ambassador to Washington Prince Khalid bin Salman.
The president declined to weigh in, once again, on the believability of the claims that Turkey says are bogus.
"We are strong and old allies".
"Yesterday (the Saudis) did not want the search to be done since the consul's family was inside", Cavusoglu said.
Members of the US Congress, including some of Trump's fellow Republicans, are among the loudest voices in the United States demanding answers and action on Khashoggi, a Washington Post columnist who moved to Washington a year ago fearing retribution for his critical views.
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The source said the timing of the two events is no accident.
In an interview with The Associated Press, Trump compared the case of Khashoggi, who Turkish officials have said was murdered in the Saudis' Istanbul consulate, to the allegations of sexual assault leveled against Supreme Court Justice Brett Kavanaugh during his confirmation hearing.
Regarding Turkey-U.S. relations after releasing Pastor Andrew Brunson, who was arrested in December 2016 over alleged ties to the Fetullah Terrorist Organization (FETO) - the group behind a defeated coup earlier that year -, Cavusoglu noted that Turkey and the U.S. are on same page regarding lifting of sanctions and other related matters.
"They said it would be a thorough, complete and transparent investigation", he said.
Bachelet stopped short of calling for an global investigation.
At least a dozen officials arrived in unmarked police cars at 6pm local time and mingled outside the building before filing inside.
Turkish President Recep Tayyip Erdogan said parts of the building had been repainted since Khashoggi, a Saudi native who criticized the crown prince's policies, vanished.
A Saudi team entered first on Monday, followed roughly an hour later by Turkish forensic police.
Searching another country's consulate - which is considered foreign soil under the Vienna Convention - is an extraordinary measure which reflects the gravity of the diplomatic crisis.
"They made a commitment to hold anyone connected to any wrongdoing that may be found accountable for that, whether they are a senior officer or official", Pompeo said.
After speaking with the king, Trump floated the idea that "rogue killers" may have been responsible for the disappearance.
"This guy has got to go", Senator Lindsey Graham told Fox. "So far, it's just a rumour of a report coming out".