Syrian regime jets risked an aerial confrontation with US warplanes on Saturday after flying over a battleground city in northern Syria despite warnings that bombing there could lead to American forces opening fire.

The US scrambled its own jets on Thursday after Assad regime aircraft dropped bombs on Kurdish forces in the city of Hasakah, dangerously close to US commandos operating on the ground.

The incident ended without shots being fired but the Pentagon warned that similar bombing raids could lead to US jets shooting down the Syrians. "The Syrian regime would be well advised not to interfere with coalition forces or its partners," a spokesman said.

It is believed to be the closest that American aircraft have come to opening fire on regime jets in the crowded skies above Syria.

Damascus appeared to be probing the limits of the American warning on Saturday as its air force flew more sorties over Hasakah, where regime troops are battling against US-backed Kurdish forces.

It was not clear whether the regime aircraft dropped bombs or not during the mission.

The rising tensions between the US and Syria came as Turkey's prime minister said his country intended to play a "more active" in the conflict in Syria.