MbS and Saudi officials continue to deny that the Crown Prince had any involvement in the killing.
Trump has also insisted that killing will not interfere with USA arms sales to Saudi Arabia, citing a $110bn deal with Riyadh he announced a year ago.
The Kremlin said in its statement on Thursday that Putin and King Salman had addressed the situation related to the "Khashoggi case" during their phone conversation.
While Saudi officials have said the crown prince, also known as MBS, had no knowledge of the plan to kill Khashoggi, this has left many unconvinced.
Khashoggi's eldest son, Salah, and his family arrived on Thursday in Washington on a flight from Saudi Arabia, according to two sources close to the family.
CIA Director Gina Haspel on Thursday briefed President Donald Trump on her "findings and discussions" in the investigation of Saudi journalist Jamal Khashoggi's death following her visit to Turkey where she reportedly listened to an audio recording of the murder, The Hill reports.
"Even Saudi Arabia itself has admitted that the crime was premeditated and involved state officials", Agnes Callamard, United Nations Special Rapporteur on Extrajudicial, Summary or Arbitrary Executions, said during a press conference.
During the vigil, a friend of Khashoggi and advisor to Turkish President Recep Tayyip Erdogan, Yasin Aktay, said that "no one can sweep this affair under the rug".
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Mourinho confirmed post-match that Chelsea boss Maurizio Sarri had ensured Ianni apologised for his role in the incident. Portuguese coach Mourinho leapt from his seat and had to be restrained by security staff as a melee ensued.
Asked by a reporter in the White House how the Khashoggi killing could have happened, Trump said: "They had a very bad original concept".
Saudi media showed Salah Khashoggi meeting Tuesday with the crown prince, who reportedly expressed his condolences.
Shortly after Khashoggi disappeared on October 2 while retrieving a document at the Saudi Consulate in Istanbul, Turkish authorities said he was the victim of a premeditated killing by 15 Saudi agents sent to Turkey on a mission to permanently silence the journalist, who had been critical of Saudi Crown Prince Mohammed bin Salman.
Turkish media have also published a security camera image allegedly showing a vehicle belonging to the Saudi Consulate "scouting" a forest in the outskirts of Istanbul before Khashoggi was killed.
However, Trump also said that he wanted to believe that rogue operatives were behind the Khashoggi killing.
An official at Turkey's presidential palace told Bloomberg, however, that while Turkey wants the crime to be uncovered, it had no interest in interfering in Saudi internal affairs or trying to influence who becomes the next king. The resolution came several days after Germany became the first Western government to suspend arms sales to Saudi Arabia, the world's largest arms importer.
For two weeks after his disappearance the Saudis had claimed that Khashoggi left the building alive.