In a phone call to Saudi Arabian Foreign Minister Adel al-Jubeir, Jeremy Hunt warned that "friendships depend on shared values".
Mr Khashoggi was last seen visiting the Saudi consulate in Istanbul last week, and Turkey says he may have been murdered there.
Saudi Arabia denies the suggestion.
Taking a tougher line than the Trump administration, a spokesperson for the UK's Foreign Office said, if media reports surrounding the case were correct, the UK would treat the incident "very seriously."
Earlier on Tuesday, US President Donald Trump said he had not yet spoken to Saudi officials about the journalist's disappearance.
"I have not. But I will be at some point," he told reporters. "I know nothing right now. I know what everybody else knows - nothing."
In a separate development, Mr Khashoggi's Turkish fiancée Hatice Cengiz appealed to the US for help.
In an emotional article in the Washington Post, she wrote: "I implore President Trump and first lady Melania Trump to help shed light on Jamal's disappearance."
"We were in the middle of making wedding plans, life plans," when he vanished, she said.
"Jamal is a valuable person, an exemplary thinker and a courageous man who has been fighting for his principles. I don't know how I can keep living if he was abducted or killed in Turkey."
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