Turkish officials had said that the journalist - a Saudi citizen and US resident - was killed and his body dismembered.
Earlier Friday, Turkey's foreign minister Mevlut Cavusoglu said his country has not provided the audio recording of Khashoggi's alleged killing to American officials but promised that Turkey would "share with the world" the results of its investigation, according to Anadolu.
Saudi state television said an investigation found that a fight broke out at the consulate between Khashoggi and people who met him at the consulate.
He dispatched US Secretary of State Mike Pompeo earlier this week to both Saudi Arabia and Turkey to speak to officials on the case.
Trump said he would be working with Congress on next moves, but "I would prefer that we don't use as retribution cancelling $110 billion worth of work, which means 600,000 jobs. we need them as a counterbalance to Iran".
An official Saudi source said on Saturday the kingdom's investigation had so far shown that the suspected killer had travelled to Istanbul to meet with Khashoggi to discuss the writer's possible return to Saudi Arabia.
It effectively dismissed Turkish claims that he had been murdered by a Saudi hit squad flown into the country for the job, and insulated Mohammed Bin Salman, the young Saudi crown prince, from blame.
The disappearance of Khashoggi, who was a USA resident and Washington Post columnist, caused tension between Saudi Arabia and Western allies, with countries including the UK, Germany and France pressing the Saudis for information on his disappearance. Canada, Russia, and Saudi Arabia, among other nations, consider the Muslim Brotherhood to be a terrorist group.
Eighteen Saudi citizens have also been detained in connection with the death.
Mr Schiff said that if Mr Khashoggi was fighting inside the consulate, he was "fighting for his life with people sent to capture or kill him".
Saturday's comments marked the first time since Khashoggi went missing that the Saudis admitted to his death.
Trump also called Saudi Arabia a "great ally".
His disappearance had been shrouded in mystery and triggered an worldwide crisis, with Turkish officials accusing Saudi Arabia of a state-sponsored killing.
Over the years, he maintained ambiguous ties with Saudi authorities, having held advisory positions in Riyadh and Washington, including to Prince Turki al-Faisal, who ran Saudi Arabia's intelligence agency for more than 20 years.
The Saudi public prosecutor didn't disclose details on the whereabouts of Khashoggi's body or who was involved in the altercation.
"God have mercy on you my love Jamal, and may you rest in Paradise", Khashoggi's fiancée Hatice Cengiz tweeted after the Saudi confirmation of the journalist's death.
Two of the sources said Saudi rulers are set to explain that Assiri had been given verbal permission from the Crown Prince to capture Khashoggi for questioning in Saudi Arabia but that he either overstepped the authorization or misunderstood his orders.