CBS News Reporter Charles Wolfson says senior officials in Washington have yet to decide on the request by Ambassador James Oberwetter.
The terror warning was only the latest in a string of alerts issued by Western governments to citizens living or visiting in Saudi Arabia, which is in a precarious position in the war on terrorism. It is a close U.S. ally, but also was home to 15 of the 19 Sept. 11 hijackers, leading some to question the Arab kingdom's commitment to fighting militants.
U.S. diplomatic missions in the Saudi kingdom are frequently closed because of threats. Last month, as part of a tightening of security at U.S. installations in the region, the embassy in Riyadh briefly closed after rumors spread of an explosion.
That closure came two days after Israel's assassination of Sheik Ahmed Yassin, founder of Hamas, prompted calls for revenge against Israeli and American targets.
State Department advisories issued after Yassin's death and still in effect said Americans face increasing threats from terror groups such as Hamas and al Qaeda.
"Credible information has indicated terrorist groups may be planning attacks against U.S. interests in the Middle East," a warning for the Middle East and North Africa read.
A worldwide caution issued last month said Washington is "deeply concerned about the heightened threat of terrorist attacks against U.S. citizens and interests abroad," as well as violent anti-American demonstrations.
It warned citizens "to maintain a high level of vigilance and to take appropriate steps to increase their security awareness."
The State Department had recommended in December that diplomats' families leave Saudi Arabia. A U.S. counterterrorism official, who spoke on condition of anonymity, said no single specific threat or piece of intelligence triggered that recommendation.
Saudi Arabia has been working to overcome that perception, as well as to put down fundamentalist opposition to the long-ruling house of Fahd.
The Saudi crackdown began after suicide bombings at three housing projects in May. Thirty-five people, including nine attackers were killed. The State Department responded then by ordering nonessential U.S. officials and family members to depart.
The diplomatic quarter east of Riyadh has been guarded heavily by Saudi armed forces since the suicide attacks. Travel by American officials and their families in Riyadh is restricted already to the hours between 6 a.m. and 6 p.m.
There are some 200 to 300 nonessential U.S. officials and family members in Saudi Arabia, and some 30,000 U.S. citizens in all.