Khashoggi entered the consulate last Tuesday and has not been heard of or seen since, according to his fiancée and friends.
Saudi officials said he left shortly afterwards but his fiancee, who was waiting outside, said he never came out.
Turkish President Recep Tayyip Erdogan is demanding that Saudi Arabia prove its claim that Khashoggi left the consulate, saying it was inadequate to simply say "he has left". He also indicated that the Saudi government is "very keen to know what happened to him".
Asked about the case while speaking with reporters outside the White House on Monday, Trump replied, "I am concerned about it".
The Foreign Office would face charges of hypocrisy if, after its outrage over the attempted assassination by Russian Federation of the former spy Sergei Skripal in Salisbury, it remained silent over an alleged murder by the Saudi government. So far, other than expressing "concern" about Khashoggi's fate, the Trump administration has said little.
"Violence against journalists worldwide is going up and is a grave threat to freedom of expression", Mr Hunt said on Twitter.
Mr Hunt stressed that if media reports from the weekend regarding Mr Khashoggi's case prove correct, that would be extremely concerning and the United Kingdom will treat the incident very seriously - friendships depend on shared values.
Aksoy said the search will take place as part of the official investigation, but did not mention when the search will be conducted.
The Vienna Convention on diplomatic relations allows for consulates to be searched by authorities of a host country with consent of the mission chief, he said.
DailyMail.com said a friend of the journalist revealed that Mr Khashoggi's encrypted messages had been read after he vanished. We're also joined by Medea Benjamin, co-founder of CodePink, who knew Khashoggi and is the author of "Kingdom of the Unjust: Behind the U.S. -Saudi Connection".
Saudi officials strongly deny any involvement in the disappearance, and insist Khashoggi left the consulate unharmed that day.
It was not immediately clear if the Anadolu report referred to one of those aircraft. It would make it more hard for the two governments to come up with a face-saving story blaming Khashoggi's disappearance on some third party, on rogue elements of the Saudi security forces, or on an accident during an interrogation that went wrong.
President Trump has said he'll talk to Saudi Arabian officials about the missing journalist at some point.
"Although my hope slowly fades away each passing day, I remain confident that Jamal is still alive".
Turkey's security services who are investigating Khashoggi's disappearance believe that within two hours of his arrival at the Saudi consulate last week, a "team of Saudi agents" killed him and then dismembered him with a bone saw that was brought in specifically for that objective.
A publication with close ties to Erdogan's government, the newspaper Sabah, reported Tuesday that unnamed officials had said the police were examining the possibility that Khashoggi had been abducted and not killed, possibly with the help of another country's intelligence officers.
United Nations spokeswoman Ravina Shamdasani said the "apparent enforced disappearance" of Khashoggi is a matter of "deep concern" to the human rights office in a press conference in Geneva on Tuesday.