Herbert, chairman of the National Governors Association, has asked the Utah Department of Public Safety to review security checks used by the federal refugee-resettlement program and wants to consult with the state's congressional delegation.
Cox left open the possibility that Herbert could change his mind after such a review.
"The highest duty of a governor," Cox said, "is to protect public safety."
The move by mostly GOP governors to reject Syrian refugees is one President Barack Obama spoke against during a news conference in Turkey, where he attended the Group of 20 summit.
The Democratic president said it was important that "we do not close our hearts to these victims of such violence and somehow start equating the issue of refugees with the issue of terrorism."
He also rejected the idea of screening refugees based on their religion, as proposed by some GOP presidential candidates, saying that is "not American."
Since the Syrian civil war began five years ago, Utah has resettled 12 Syrians, from two families, the most recent of whom arrived eight months ago. The state expects to receive a few hundred more between March and October.
That group would be part of Obama's plan to boost the total number of refugees accepted in the next year from 70,000 to 85,000. He said at least 10,000 of them would be Syrians.
In recent months, hundreds of thousands of Syrians have flooded Europe, where the security checks have been limited to a review of international arrest warrants.
Authorities in Paris found a Syrian passport near one of the eight attackers, who killed nearly 130 people in four coordinated attacks Friday. That passport was scanned in Greece, Serbia and Croatia in October, according to The Associated Press. It is unclear whether it is a real passport or whether it belongs to the suicide bomber. The militant Islamic State group has claimed responsibility for the attacks.
A bloc of Republican governors in the United States says the connection is reason enough to deny Syrian refugees access. So far, the chief executives in Alabama, Arizona, Arkansas, Florida, Georgia, Idaho, Illinois, Iowa, Kansas, Louisiana, Maine, Massachusetts, Michigan, Mississippi, Nebraska, Nevada, New Hampshire, New Jersey, New Mexico, North Carolina, Ohio, Oklahoma, South Carolina, Tennessee, Texas and Wisconsin say they do not want to admit any more Syrians, according to a tally kept by CNN. New Hampshire is the only state on that list led by a Democrat.
Texas Gov. Greg Abbott sent a letter to Obama expressing doubt that the federal government can perform the necessary security checks.
"Neither you nor any federal official can guarantee that Syrian refugees will not be part of any terroristic activity," Abbott said. "As such, opening our door to them irresponsibly exposes our fellow Americans to unacceptable peril."
Michigan Gov. Rick Snyder explained his decision to postpone accepting refugees by saying his "first priority is protecting the safety of our residents."