A source quoted by The Washington Post said the journalist was killed by a 15-member Saudi team sent "specifically for the murder".
The sources did not say how they believed the killing was carried out.
Prince Mohammed said in an interview published by Bloomberg on Friday that the journalist had left the consulate and Turkish authorities could search the building, which is Saudi sovereign territory.
The two unnamed sources said on Saturday that the initial assessment of police was that Mr Khashoggi had been killed at the consulate.
Early Sunday, official Saudi Press Agency released a statement saying that an unidentified official at the Saudi Consulate in Istanbul had dismissed the reports that Khashoggi was killed there. Another official confirmed the story.
Lebanon's Al Akhbar newspaper cited an Arab source as saying Saudi Arabia had informed Turkey that Khashoggi was now in the Saudi capital Riyadh. He is also a columnist with The Washington Post and has criticized some of the Saudi government's policies, in addition to supporting the Muslim Brotherhood. Ankara says there has been no evidence to corroborate the claim.
The journalist has lived in self-imposed exile in Washington for the a year ago, fearing retribution for his views on Saudi policies.
"The condition of the lost journalist, details on him and who is responsible for this will be uncovered", AK Party spokesman Omer Celik told reporters at a party summit chaired by Turkish President Recep Tayyip Erdogan.
Khashoggi's disappearance has drawn attention to Prince Salman's crackdowns on his critics.
Meanwhile, Saudi Crown Prince Mohammed bin Salman denied speculations that Riyadh had taken Khashoggi, stressing that Turkey could search the kingdom's Consulate in Istanbul for the journalis.
"Most of their cases have nothing to do with freedom of speech and majority will return to their homes when the process is finished", he said.
"We will allow them to enter and search and do whatever they want to do". But he added that diplomatic cars had been seen moving in and out. 'We have nothing to hide'.
Khashoggi is a familiar face on political talk shows on Arab satellite television networks and used to advise Prince Turki al-Faisal, former Saudi intelligence chief and ambassador to the United States and Britain.
The New York-based Committee to Protect Journalists demanded Riyadh give "a full and credible account" of what happened to Khashoggi inside the consulate.
Protestors hold pictures of missing journalist Jamal Khashoggi during a demonstration in front of the Saudi Arabian consulate in Istanbul on October 5, 2018.
Khashoggi fled from Saudi Arabia in September 2017, months after Prince Mohammed was appointed heir to the throne, amid a campaign that saw dozens of dissidents arrested including intellectuals and Islamic preachers.
The Turkish police in Istanbul has been keeping an eye on the entry and exits at the Saudi Consulate since the time, Khashoggi entered the building, his fiancée said.