WASHINGTON—Frustrated by the stalemate in Syria, Secretary of State John Kerry has been pushing for the U.S. military to be more aggressive in supporting the country's rebel forces. Opposition has come from the institution that would spearhead any such effort: the Pentagon.

Mr. Kerry and United Nations Ambassador Samantha Power have advocated options that range from an American military intervention to weaken the regime of President Bashar al-Assad to using U.S. special operations forces to train and equip a large number of...