The fiancee of the Saudi journalist, Hatice Cengiz told Turkish TV she thought the invitation was aimed at influencing public opinion in the US.
Khashoggi was killed in the Saudi consulate in Istanbul three weeks ago.
Riyadh denies the ruling royal family was involved and blames “rogue agents”.
Saudi Arabia initially denied all knowledge of the journalist’s fate but the Saudi public prosecutor now describes it as premeditated murder.
Mr Trump has said he is “not satisfied” with the Saudi account. However, while he has floated the possibility of imposing sanctions. he has also stressed the importance of the two countries’ ties.
The president also said it was “possible” Crown Prince Mohammed bin Salman did not know about the killing.
In comments made on Saturday at security conference in Bahrain, US Defence Secretary Jim Mattis said Mr Khashoggi’s death in a diplomatic facility “must concern us all”.
He said the United States “does not tolerate this kind of ruthless action to silence Khashoggi, a journalist, through violence.”
“Failure of any nation to adhere to international norms and the rule of law undermines regional stability at a time when it is needed most,” he added.
On Friday, Turkey said it wanted to extradite 18 Saudis who were arrested in Riyadh in connection with the killing.
Prosecutors preparing the formal request accuse them of “murder by premeditation, monstrous intent or by torture”.
Turkey and Saudi Arabia are not known to have an extradition treaty.
What did his fiancée say?
In a tearful television interview on Friday, Ms Cengiz recounted the day her fiancé vanished, saying she would never have let Khashoggi enter the consulate if she had thought that the “Saudi Arabia authorities would hatch a plot” to kill him.
“I demand that all those involved in this savagery from the highest to the lowest levels are punished and brought to justice,” she told Haberturk TV.
She said she had not been contacted by any Saudi officials but that she was unlikely to go to Saudi Arabia for a funeral, should Khashoggi’s missing body eventually be found.
Turkish President Recep Tayyip Erdogan has called on Riyadh to admit who ordered the killing and where the body is.
But there is disagreement over how to respond.
While Germany has suspended arms exports to the Gulf nation, French President Emmanuel Macron said it would be “pure demagoguery to say ‘we must stop arms sales'”.
The sale of weapons “has nothing to do with Khashoggi. One shouldn’t mix everything up”, he said on Friday.
The European Parliament on Thursday passed a non-binding resolution urging an EU-wide arms embargo on Saudi Arabia in response to the murder.
Russia has said the royal family should be believed and that “no-one should have any reasons not to believe them,” Kremlin spokesman Dmitry Peskov said.