Saudi authorities dismissed the allegation as "baseless".
Jamal Khashoggi was last seen visiting the Saudi consulate last Tuesday.
Saudi Arabia's Mohammed bin Salman has implemented a string of reforms in his country, but with his ascension to crown prince in June 2017 has come an intensified crackdown on dissent.
'If the reports of Jamal's murder are true, it is a monstrous and unfathomable act, ' the Washington Post's editorial page editor Fred Hiatt said in a statement. "We've tried to push to IPO it as soon as possible, but this is the timing, based on the situation that we have".
A Turkish official, requesting anonymity to discuss the ongoing investigation, told the AP earlier Saturday night something similar.
But Turkish police believe he was killed by a team of assassins who were sent to Istanbul and departed the same day, according to a government source. "We believe that the murder was premeditated and the body was subsequently moved out of the consulate".
Khashoggi had gone to the consulate to obtain documents needed to marry his Turkish fiancee. The consulate insists the writer left its premises, contradicting Turkish officials.
'Jamal is not dead!
Several Turkish police sources, speaking on the condition of anonymity, have told news agencies that Khashoggi was likely killed inside the building. It added the investigation over allegations that the writer was detained had "deepened", without elaborating.
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Prince Mohammed said he loved working with Trump. "It's part of the sovereignty issues that are stuck, unsolved, between Saudi Arabia and Kuwait for the past 50 years".
The string of arrests and the Hariri case have reflected poorly on the image of Prince Mohammed as a "reformer".
Khashoggi was known for his interviews and travels with Osama bin Laden between 1987 and 1995, including in Afghanistan, where he wrote about the battle against the Soviet occupation. He said, last week, that it plans to raise a new Dollars 100 billion fund every two or three years.
Saudi Arabia and allies invaded Yemen in March 2015 to reinstate Riyadh-allied former officials.
Khashoggi, who has been missing since he entered the Saudi consulate in Istanbul on October 2, had been banned from writing in the pan-Arab Al-Hayat newspaper following his defence of the Muslim Brotherhood, which Riyadh has blacklisted as a terror organisation. The consulate has also denied that Khashoggi was abducted.
"If this is true - that the Saudis lured a USA resident into their consulate and murdered him - it should represent a fundamental break in our relationship with Saudi Arabia", Murphy wrote on Twitter.
Media freedom organizations and human rights groups have demanded information on his whereabouts.
The president also issued a warning to the Saudi consulate, saying that they can not save themselves just by saying that Khashoggi left the building, and that they must provide concrete evidence about the whereabouts of the missing journalist and prove that he exited the premises. "If the killing is confirmed, those responsible for this heinous crime must be held accountable", she added.