ISTANBUL—Turkey is set to launch a military intervention in northeastern Syria to combat U.S.-backed Kurdish rebels it regards as terrorists, drawing a stern warning from the Pentagon and signaling new tension between the two North Atlantic Treaty Organization allies.

“We will start our operation in a few days to liberate areas east of the Euphrates River from terrorist organizations,” President Recep Tayyip Erdogan said on Wednesday in a speech at the presidential palace in Ankara.

Mr....