Khashoggi, a critic of de facto Saudi ruler Prince Mohammed bin Salman, was killed in Saudi Arabia's Istanbul consulate last month in a hit which Erdogan says was ordered at the "highest levels" of the Saudi government.
Saudi Arabia's public prosecutor said he was seeking the death penalty for five out of 11 suspects charged in the murder of the prominent journalist.
The dissident writer was given a lethal injection after a struggle with agents in the Saudi consulate in Istanbul on 2 October, a spokesman told reporters.
An overdose, however, led to his death, the statement said, adding that the body was then dismembered and removed by five people from the consulate.
According to the Times, one of Khashoggi's assassins made a phone call to his superior shortly after the murder.
However, it appears al-Mojeb has stopped short of accusing al-Assiri of ordering the killing itself - further distancing the killers from the crown prince's inner circle.
In his Thursday statement, the Saudi prosecutor also said the country had detained a total of 21 people over the killing.More news: Ranking the 10 best players acquired by the Philadelphia 76ers via trade
The prosecutor also said the investigation has identified the individual who wore the victim's clothes after the murder and disposed of the victim's belongings in a trash receptacle, including his watch and glasses, after leaving the consulate building.
Meanwhile, Turkish Foreign Minister Mevlut Cavusoglu said Thursday that the Saudi actions were "positive but insufficient".
Shaalan also went on to say that Saudi investigators had requested that Turkey submit all original recordings and evidence related to the case, but that it has yet to fulfil that request.
Khashoggi had been living in self-imposed exile in the United States since past year. "We have to share with the worldwide community whatever Turkey has in its hands", Cavusoglu said in Ankara.
He added that Saud al-Qahtani, a former advisor to Crown Prince Mohammed bin Salman, is also under investigation and barred from traveling overseas. "We played them to all who wanted them including the Saudis, the USA, France, Canada, Germany, Britain", he said.
Saudi authorities initially stated the journalist left the consulate, before backtracking and admitting on October 20 he was killed by "rogue" operatives.