The move by the State Department signifies that American officials recognize the growing risks in the region, the publication said.
According to the newspaper, the State Department order will apply not only to US government employees in Riyadh but also in Jeddah and Dhahran, two cities where US consulates are located.
Earlier, American diplomats criticized the Trump administration for delaying preparations for evacuating the Middle East, despite weeks of US troop buildup in the region and openly threatening Iran with war.
On February 28, the US and Israel launched an unprovoked aggression against Iran, including Tehran, causing damage and civilian casualties. Iran is carrying out retaliatory strikes on Israeli territory, as well as on US military targets in the Middle East.