Saudi Arabia says Crown Prince Mohammed bin Salman spoke Monday by telephone with the son of slain journalist Jamal Khashoggi to express condolences for the killing.
Khashoggi died after entering the Saudi consulate in Istanbul, Turkey on October 2.
Turkish President Recep Tayyip Erdogan is vowing to reveal details about the case in a Tuesday speech to his parliament.
He told an Istanbul rally Sunday, "We are looking for justice here and this will be revealed in all its naked truth, not through some ordinary steps."
Erdogan spoke to U.S. President Donald Trump by telephone Sunday. Turkey's state-run news agency said both leaders agree the Khashoggi case needs to be "cleared up with all aspects."
Saudi Arabia called Khashoggi's killing inside its Istanbul consulate "a huge and grave mistake" and vowed those responsible for it would be held accountable.
Foreign Minister Adel al-Jubeir told Fox News Sunday that Saudi agents "did this out of the scope of their authority," calling it "a rogue operation."
The top Saudi diplomat offered his condolences to Khashoggi's family, but disclosed no new information about how the writer was killed, where his body is or if Crown Prince Mohammed bin Salman -- the country's de facto ruler - was involved.
"There obviously was a tremendous mistake made and what compounded the mistake was the attempt to try to cover up," al-Jubeir said. "That is unacceptable in any government."
Saudi Arabia claims the 59-year-old Khashoggi was killed October 2 after an argument leading to a fist fight -- an explanation that has drawn widespread international scorn and skepticism, including from Trump. After he initially seemed willing to believe Saudi accounts, the president now says "obviously there has been deception, and there has been lies."
Al-Jubeir said in the Fox television interview, "This is an aberration. This is a mistake and those responsible will be punished for it. We want to make sure that we know what happened and we want to make sure that those responsible be held to account." Saudi Arabia says it has fired five key officials linked to the death and arrested 18 others.
Critics are questioning how a team of 15 Saudi agents could fly to Istanbul to meet Khashoggi and eventually kill him without the crown prince's knowledge and consent. But al-Jubeir said, "There were not people closely tied to him," although news accounts have said that several Saudi security officials close to Mohammed were involved.
Khashoggi was living in the U.S. in self-imposed exile, writing columns for The Washington Post that were critical of Mohammed and Saudi Arabia's intervention in the conflict in Yemen.
Trump told the Post that Saudi Arabia has been an "incredible ally" of the United States for decades and it is possible the crown prince did not order Saudi agents to kill Khashoggi.
"Nobody has told me he is responsible. Nobody has told me he is not responsible," the U.S. leader said. "We have not reached that point...I would love if he was not responsible."
Numerous U.S. lawmakers, including Trump's Republican colleagues, are calling for sanctions against the Saudis. Turkish investigators say Saudi agents tortured Khashoggi, decapitated him and then dismembered his body.
Trump told the Post that "something will take place" in response to Khashoggi's death, but said the United States should not let the incident disrupt a possible $110 billion weapons sale to Riyadh he announced last year.
"It's the largest order in history," Trump said. "To give that up would hurt us far more than it hurts them. Then all they'll do is go to Russia or go to China. All that's doing is hurting us."
But one Trump supporter, Kentucky Senator Rand Paul, told Fox "I don't think arms should ever be seen as a jobs program."
Other U.S. lawmakers voiced skepticism of the Saudi explanation for Khashoggi's death.
Senator Bob Corker, chairman of the Senate Foreign Relations Committee, told CNN he believes Mohammed bin Salman was responsible, saying, "Yes, I think he did it."
A Trump critic, Democratic California Congressman Adam Schiff, told ABC News, "This ought to be a relationship-altering event for the U.S. and Saudi Arabia that we ought to suspend military sales, we ought to suspend certain security assistance."
U.S. officials are faced with reconciling the Saudi explanation for Khashoggi's death and Turkey's claim an audio recording exists of Khashoggi's torture and death. Trump denies U.S. officials have heard the audio or read transcripts of it, but the Post quoted sources saying that Central Intelligence Agency officials have listened to the audio. Verification of it would make it difficult to accept the Saudi explanation for Khashoggi's death.
European leaders and the human rights group Amnesty International expressed skepticism about the Saudi explanation.
Britain, Germany and France issued a joint statement condemning the killing of Khashoggi and said there is an "urgent need for clarification of exactly what happened." They said the Saudi explanation for the journalist's death needs to be supported by facts in order to be credible.
European Union foreign policy chief Federica Mogherini said the circumstances around Khashoggi's death are deeply troubling, and called for a thorough, credible and transparent investigation.
Amnesty International called on Saudi Arabia to "immediately produce" Khashoggi's body so an autopsy can be performed.
Amnesty's director of campaigns for the Middle East, Samah Hadid, said a United Nations investigation would be necessary to avoid a "Saudi whitewash" of the circumstances surrounding Khashoggi's death. Hadid said such a cover-up may have been done to preserve Saudi Arabia's international business ties.
