ISTANBUL—A U.S.-backed opposition faction is cooperating with an al Qaeda-linked rebel group to battle the Syrian regime as the opposition chief prepares to meet with officials in Washington to lobby for weapons, in part to fight al Qaeda.

The U.S. supports the moderate and secular-leaning Free Syrian Army. But Washington refuses to supply the rebel group with sophisticated weaponry for fear those arms will fall into the hands of al Qaeda-linked extremists fighting the regime alongside the FSA and myriad other factions in...