In another escalation of the five-year war, regime planes this week bombarded positions held by US-backed Kurdish forces in the city fighting the Islamic State (IS) jihadist group.
Turkish Prime Minister Binali Yildirim said the raids showed Damascus was starting to see the Kurdish attempt to consolidate territory in northern Syria as "a threat".
He pledged to play a "more active" in the next months in putting an end to the conflict.
The unprecedented strikes prompted the US-led coalition to scramble aircraft to protect its special operations forces helping the Kurdish fighters.
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It was apparently the first time the coalition scrambled jets in response to regime action, and possibly the closest call yet in terms of Syrian forces wounding American or coalition advisers.
Regime warplanes were in the air above Hasakeh throughout the night and into the morning today, according to the Syrian Observatory for Human Rights, a UK-based monitoring group.
Fighting erupted between pro-government militia and the US-backed Kurdish forces on Wednesday and continued into the morning today, leaving a total of 41 people dead including 25 civilians, the Observatory said.
A delegation of Russian officials from the coastal Hmeimim military airport arrived in Qamishli to the north to hold talks between the two sides, a senior Syrian government source told AFP.
A journalist in Hasakeh said this afternoon that the clashes had abated.
Around two-thirds of the city is controlled by Kurdish forces, while the rest is held by pro-government militia.
The vitriol between the two sides escalated today, as the Kurdish People's Protection Units (YPG) pledged to "protect areas from the terrorism of the regime.
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