Salah Khashoggi had previously been unable to leave Saudi Arabia due to his father's criticism of the leadership.
Jamal Khashoggi was murdered in the Saudi consulate in Istanbul three weeks ago, investigators say.
Saudi Arabia's public prosecutor is travelling to Turkey on Sunday as part of the investigation.
Turkish President Recep Tayyip Erdogan announced the move in a televised speech, adding that the prosecutor would meet Istanbul's prosecuting authorities.
Turkey also had other evidence, which had not yet been disclosed, Mr Erdogan said.
And he called on Saudi Arabia to disclose the location of Jamal Khashoggi's body, as well as the name of an alleged "local co-operator" involved in disposing of it.
The Saudi prosecutor, Sheikh Saud al-Mojeb, said on Thursday that the killing was premeditated, in a further development of the Saudi narrative.
Saudi officials have also blamed "rogue" agents as Riyadh has struggled to provide a credible explanation for how Jamal Khashoggi met his end.