The Syrian government in Damascus has offered no confirmation.
Turkey regards the Kurdish fighters, just across its border in Afrin, as terrorists. It launched a major offensive against them last month.
There is currently no Syrian military presence in the area.
A senior Kurdish official, Badran Jia Kurd, told Reuters that government soldiers could enter the Afrin region within days and that they would deploy to some border positions.
The alleged agreement was also reported by Iraqi Kurdish media group Rudaw, which quoted a Kurdish politician from Syria, and a news agency which backs Syrian Kurdish forces.
If the deal has really been struck, Turkish troops could find themselves confronting not only Kurdish fighters in Afrin, but the Syrian army too, says BBC World Service Middle East editor Alan Johnston.
Syrian President Bashar al-Assad’s soldiers withdrew from northern Kurdish areas in 2012.
The Kurds of the Democratic Union Party (PYD) – the dominant Kurdish party – then quickly took charge, backed by its armed wing, the People’s Protection Units (YPG).
The YPG cleared Islamic State (IS) group fighters from wide swathes of Syria.
Turkey is trying to oust the YPG from Afrin because it sees the group as an extension of the banned Kurdistan Workers’ Party (PKK), which has fought for Kurdish autonomy in Turkey for three decades.
The YPG denies any direct military or political links with the PKK.
The Syrian military and the YPG have largely avoided direct conflict in the Syrian war thus far but have had sporadic clashes.
“We can co-operate with any side that lends us a helping hand in light of the barbaric crimes and the international silence,” Mr Jia Kurd said.
However, he added that the alleged agreement – which he said did not include any political arrangements – could fall through.
“We don’t know to what extent these understandings will last because there are sides that are not satisfied and want to make [it] fail,” he said.
Northern and north-western Syria hosts an extremely complex mix of alliances between different international and regional players.
According to Reuters, a Kurdish political official said Russia could object to any agreement between the YPG and the Syrian government as it complicates its own diplomatic efforts with Turkey.
The situation is further complicated by the YPG militia receiving funding from the United States, as allies in the fight against IS. (Courtesy BBC)