The latest reported fighting comes despite the United States saying Wednesday that it was working to address Turkey's concerns in Syria to dissuade the NATO ally from escalating an offensive against Kurdish fighters.

The Syrian Observatory for Human Rights war monitor reported "fierce battles in the Manbij countryside... in the past hours between the (Kurdish-led) Syrian Democratic Forces and the (Turkish-backed) National Army factions... with Turkish air cover".

"The attacks killed 37 people in a preliminary toll," mostly Turkish-backed combatants, but also six SDF fighters and five civilians, said the British-based Observatory with a network of sources inside Syria.

The monitor said at least 322 people have been killed in fighting in the Manbij countryside since last month.

On Wednesday, Mazloum Abdi, who heads the US-backed SDF, said his group supported "the unity and integrity of Syrian territory". In a written statement to AFP, he called on Syria's new authorities "to intervene in order for there to be a ceasefire throughout Syria".

Abdi's comments followed what he called a "positive" meeting between Kurdish leaders and the Damascus authorities late last month.

Turkish-backed factions in northern Syria resumed their fight with the SDF at the same time as Islamist-led rebels were launching an offensive on November 27 that overthrew Syrian president Bashar al-Assad just 11 days later.

The pro-Ankara groups succeeded in capturing Kurdish-held Manbij and Tal Rifaat in northern Aleppo province, despite US-led efforts to establish a truce in the Manbij area.

- Mounting casualties -

The fighting has continued since, with mounting casualties.

On Wednesday Washington's Secretary of State Antony Blinken said Turkey had "legitimate concerns" about Kurdistan Workers' Party (PKK) militants inside Syria and called for a resolution in the country that includes the departure of "foreign terrorist fighters".

"That's a process that's going to take some time, and in the meantime, what is profoundly not in the interest of everything positive we see happening in Syria would be a conflict, and we'll work very hard to make sure that that doesn't happen," Blinken told reporters in Paris.

Turkey on Tuesday threatened a military operation against Kurdish forces in Syria unless they accepted Ankara's conditions for a "bloodless" transition after Assad's fall.

Syria's Kurds control much of the oil-rich northeast of the country, where they enjoyed de facto autonomy during much of the civil war since 2011.

The US-backed SDF spearheaded the military campaign that ousted Islamic State group jihadists from their last territory in Syria in 2019.

But Turkey accuses the main component of the SDF, the People's Protection Units (YPG), of being affiliated with the PKK, which has waged a four-decade insurgency against the Turkish state.

The PKK is considered a terrorist organisation by Turkey, the United States, the European Union and most of Turkey's Western allies.

Turkey has mounted multiple operations against the SDF since 2016.