SALT LAKE CITY—Amid concerns over potential terrorists infiltrating the U.S., Gov. Gary Herbert of Utah has repeatedly pledged to keep the welcome sign hanging for refugees while most of his Republican counterparts want a review or pause in resettlement.

After the San Bernardino, Calif., and Paris attacks, Mr. Herbert met with national-security officials to better understand the refugee vetting process. Utah police, he said, have begun working directly with the state’s refugee communities to nurture ties with them and provide...