Turkish officials have said he was killed on the premises and his body removed.
On Tuesday, Jeremy Hunt met the Saudi ambassador and spoke by telephone to his counterpart in the state to voice the UK's concerns about the case.
Mr Hunt said that if reports of Mr Khashoggi's death proved correct, the UK would regard the situation as "serious", adding: "Friendships depend on shared values."
Mr Hunt urged the Saudi Government to co-operate fully with the Turkish investigation into the case, and to provide further information as soon as possible.
The Turkish authorities are set to conduct a search of the Istanbul consulate building after announcing that Saudi Arabia had declared itself "open to co-operation" in the investigation.
Turkish President Recep Tayyip Erdogan has not confirmed the alleged killing, saying he would await the result of an investigation.
Saudi officials have denied the allegations as baseless.