He added that Saud al-Qahtani, a former advisor to Crown Prince Mohammed bin Salman, is also under investigation and barred from traveling overseas.
Jamal Khashoggi was murdered at the Saudi consulate in Istanbul on October 2.
Saud al-Mojeb told a news conference in Riyadh on Thursday that five men ordered drugging and dismemberment of Khashoggi after "talks with him failed" inside the country's consulate in Istanbul.
A spokesman for the prosecutor later denied that Salman had any knowledge of the killing.
The spokesman said Mr al Assiri had given an order to force Mr Khashoggi home and that "the head of the negotiating team" that flew to Istanbul had ordered his murder. The Crown Prince has blamed it on "rogue elements" of the Saudi state.
Saudi prosecutors said al-Assiri deemed Khashoggi a threat because of his work as a writer and he was allegedly backed by groups and countries that are hostile to Saudi Arabia.
His body was then dismembered and taken out of the building, the Al-Mojeb, though admitted the kingdom does not not know where the remains are now.
Khashoggi's killing has plunged Saudi Arabia into its worst diplomatic crisis since the September 11 attacks.
"The Public Prosecutor has requested the death penalty for 5 individuals who are charged with ordering and committing the crime and for the appropriate sentences for the other indicted individuals", Shaalan said, without naming the five.
Their cases have been referred to a court while investigations into another 10 people suspected of involvement continue.
On Wednesday, Turkish Foreign Minister Mevlut Cavusoglu called for an global investigation into the case.
"The leader of the mission to repatriate Khashoggi chose to kill him when he failed to convince him to return", Shaalan said.
Khashoggi had been living in self-imposed exile in the United States since previous year.