The ministers from four counties gave the governor's office a petition they said had 1,700 names on it asking the governor to call a special session of the state legislature so lawmakers could pass a law exempting Davis and others from issuing marriage licenses.
Beshear cannot run for re-election because of term limits. Beshear's chief of staff told the pastors the governor would not call a special session because it would cost too much money for an issue that only affects a few county officials.
Tuesday's meeting comes three weeks before Kentucky's election for governor.