Representatives for the White House did not immediately respond to a request for comment on the letter, which was first reported by NBC News.
Erdogan maintains that a 15-strong team traveled from Riyadh to kill Khashoggi, who had been a vocal critic of Saudi Arabia's Crown Prince and de facto ruler Mohammad bin Salman, at the Saudi diplomatic compound in Istanbul.
Saudi officials have said the kingdom will try the 18 and bring them to justice after the investigation is complete.More news: The Google Home Page Now Has Discover Feed
Fidan said that talks with Saudi Arabia had produced "no concrete results".
Mr Khashoggi, a 59-year-old columnist for The Washington Post, vanished after entering the consulate in Istanbul to pick up paperwork for his upcoming marriage to his fiancee, who was waiting for him outside.
The spokesman for Turkey's ruling AK Party, Omer Celik, said Khashoggi's killing could not have been carried out without orders from someone in a senior position in the Saudi Government. The kingdom has faced intensifying global pressure to be transparent about the death of Khashoggi, a columnist for The Washington Post who was a critic of Saudi Crown Prince Mohammed bin Salman.
Over the period - which was either side of the last general election - Saudi Arabia was the government that spent the most on visits by MPs, followed by Taiwan and Hong Kong.More news: Judge blocks Georgia from rejecting absentee ballots due to mismatched signatures
Lawmakers led by Marco Rubio of Florida said in a letter to the president that they had reservations about the transfer of nuclear power technology even before the October 2 slaying of Khashoggi at the Saudi Consulate in Istanbul.
The statement from chief prosecutor Irfan Fidan comes after weeks of reports in local media about the murder at the Saudi consulate in Istanbul.
Al-Mojeb was the first Saudi official to acknowledge that the killing was "premeditated" based on the results of Turkey's investigation. And Turkey wants to know who ordered the journalist's slaying.
"There is no point in procrastinating or trying to save some people from under this,"he told reporters on Tuesday".
Cengiz had tough words for both Saudi Arabia and the United States.More news: Mueller refers plot to make false claims about him to FBI
"The president of the United States has to look at this from the point of view of humanity and consider it an worldwide tragedy, and it should come before global politics and diplomacy", she said.