The two sides have mostly avoided confrontation during the five-year conflict, with the government focusing its efforts against Sunni Arab rebels in the west, and the Kurds mainly fighting Islamic State in northern Syria.
In an indication of their reluctance to escalate further, pro-government media said on Saturday they had held preliminary peace talks.
After the fighting broke out this week, government warplanes bombed Kurdish-held areas of Hasaka, one of two cities in the largely Kurdish-held northeast where the government has maintained enclaves.
Fighting there could complicate the battle against Islamic State because of the Kurds’ pivotal role in the U.S.-backed Syrian Democratic Forces’ (SDF) fight against the group.
On Friday, warplanes from the U.S.-led coalition flew what the Pentagon called protective patrols around Hasaka to prevent Syrian jets from targeting U.S. special forces, who are operating on the ground with the SDF, the first sorties of their kind in the war.
Ground fighting intensified late on Friday when Kurdish YPG fighters battled Syrian forces, whose air force flew sorties over the city, Kurds and monitors said.
“The clashes continue in areas inside the city today. There were military operations,” a Kurdish official said.
Many inhabitants of Kurdish areas fled on Friday and at least 41 people have been killed, the Syrian Observatory for Human Rights, a U.K.-based war monitoring group, said.
“There are efforts to cool things between the army and the Asayish (YPG-affiliated forces), and a first meeting was held aimed at a ceasefire,” Sham FM, a pro-government radio station, reported.
As well as complicating the war against Islamic State, fighting in Hasaka could create problems for the government’s campaign in the city of Aleppo, where Kurdish forces have been accused of coordinating with the Syrian army against rebels backed by Turkey.
The YPG, or People’s Protection Units, have close ties with the Kurdistan Workers Party (PKK) in Turkey, against which Ankara has waged a three-decade counter insurgency. Turkey fears the Kurds’ drive against Islamic State is partly aimed at carving out a Kurdish region along its own southern border.
On Saturday, Prime Minister Binali Yildirim said Turkey would take a more active role in Syria in coming months to stop it being torn along ethnic lines – an apparent reference to the YPG gains in northern areas.
Local fighters backed by the SDF, of which the YPG militia form an integral part, said on Saturday they would not advance further north – towards the Turkish border – having secured the city of Manbij, 250 km (155 miles) west of Hasaka, from Islamic State, an announcement that may have been aimed at assuaging Turkish fears.
Syria’s army has blamed the YPG for the Hasaka fighting and described it as a branch of the PKK, a characterization the group rejected on Saturday.
In Aleppo, fighting continued near the mouth of a corridor that rebels opened this month into besieged areas they control.
Jakob Kern, the Syria director of the United Nations’ World Food Programme, said opposition-held areas had been inaccessible for weeks and food was running perilously short.
“In the east of Aleppo, the food will last a maximum of two weeks, probably until the end of August,” Swiss newspaper Tages-Anzeiger quoted him as saying on Saturday.
Russia, the main military backer of Syrian President Bashar al-Assad, said on Thursday it was willing to support weekly 48-hour ceasefires to allow aid to reach besieged areas.