The six-day campaign against the Syrian Kurdish People's Protection Units (YPG) has seen Washington's fellow NATO member Ankara attacking a US-allied force, even raising fears of military confrontation between two alliance powers.

After the Turkish and US presidents spoke late on Wednesday, the White House said Trump had urged Turkey "to de-escalate, limit its military actions", expressing concern that the assault could harm the fight against jihadists.

But a Turkish official said the US statement did "not accurately reflect the content" of the call.

Turkish Foreign Minister Mevlut Cavusoglu even suggested that the US statement had been concocted in advance of the talks.

On his first visit to troops in the southern Turkish border region of Hatay since the campaign began, Erdogan emphasised that operation Olive Branch "will continue until the result is reached".

Turkey launched an offensive against the YPG militia on Saturday in their enclave of Afrin, supporting Syrian rebels with air strikes and ground troops.

The local authority in Afrin on Thursday called on Syria to intervene to stop Turkish warplanes flying overheard.

Damascus should "say that it will not allow Turkish planes to fly in Syrian airspace," said Othman Al-Sheikh Issa, co-chair of Afrin's executive council.

- 'Continue east of the Euphrates' -

Ankara views the YPG as a terror group linked to the outlawed Kurdistan Workers' Party (PKK) inside Turkey, which is blacklisted by Ankara and its Western allies.

But the YPG is still working closely with Washington against the Islamic State (IS) extremist group in Syria.

Quoted by the presidency, Erdogan said that once the "cleansing of terrorists" is completed in Afrin, the area will be handed back to its "principal residents".

"Turkey does not have its eye on another country's territory," he was quoted as saying.

Further raising the stakes, Ankara has also floated an operation on Manbij, a YPG-held town to the east where there is a US military presence, and even to areas east of the Euphrates up to the Iraqi border.

"The second phase will be Manbij and after that, the east of the Euphrates," Erdogan's senior adviser Ilnur Cevik said.

With Europe also concerned, Germany has asked for talks at NATO over the campaign and also put a temporary halt on arms deals with Turkey, Foreign Minister Sigmar Gabriel said.

NATO chief Jens Stoltenberg said Turkey had a right to defend itself but emphasised "this has to be done in a proportionate and measured way".

Turkish Prime Minister Binali Yildirim hit out at the US, saying "the country we call an 'ally' in NATO is in cahoots with terror organisations".

Deputy Prime Minister Bekir Bozdag said that if the US wanted to "avoid a confrontation with Turkey -- which neither they nor Turkey want -- the way to this is clear: they must cut support given to terrorists".

- 'Risks giving life to IS' -

The US envoy to the coalition against IS, Brett McGurk, said on Twitter the "prolonged operation risks giving life to ISIS (IS) as it's on verge of defeat".

Washington has more than 2,000 special forces and support troops inside Syria, mainly east of the Euphrates in an area also controlled by the YPG but separate from Afrin, which is west of the river.

A Pentagon spokesman said US and Turkish military commanders had in the past years discussed the possibility of creating a "secure zone" along the border with Syria, but no decision was made.

An AFP correspondent saw tanks on the Turkish side of the border and soldiers ready to go into Syria amid tight security. Turkish artillery fire pounded the Afrin region, state-run news agency Anadolu said.

A Turk and a Syrian were killed Wednesday by YPG rocket fire on Kilis, its governor said. The YPG has denied shelling Turkish border towns like Kilis.

Hundreds attended their funeral on Thursday.

Three Turkish soldiers have been killed since the start of the offensive, while the Syrian Observatory for Human Rights has said 52 Ankara-backed Syrian rebels and 47 US-backed Syrian Democratic Forces and YPG fighters had been killed.

The SDF is an umbrella grouping composed mainly of YPG.

The Turkish military said at least 306 "terror organisation members were neutralised" but it was not possible to verify this.

The Observatory has said at least 30 civilians have been killed but Ankara strongly rejects such claims, saying it is doing everything to avoid civilian casualties.