Two boys walk by Turkish army tanks in Afrin, Syria, on Jan. 22, 2018. Turkey launched on Jan. 20 a new operation dubbed as "Operation Olive Branch" on the ground to oust Kurdish militia from Syria's Afrin. (Xinhua photo)

ISTANBUL, Jan. 26 (Xinhua) -- As the Turkish military advances in Syria's Afrin against Kurdish militia forces, the U.S. call for a limitation of the operation's scope reveals rising tension in ties between the two NATO allies.

Hours after a phone call between Turkish President Recep Tayyip Erdogan and his U.S. counterpart Donald Trump, Turkish jets struck a group of Kurdish militia in rural Manbij, local media reported Wednesday.

Some U.S. troops are deployed in Manbij, which is also under the control of the Kurdish militia known as the People's Protection Units (YPG).

"Risk is high for Turkish army to directly clash with the U.S. troops in Manbij," Haldun Solmazturk, a former general of the Turkish military, told Xinhua.

Ankara launched last Saturday a military operation, dubbed Olive Branch, in a bid to push the Kurdish militia out of Afrin on its border.

A statement issued by the White House following Wednesday's phone call said Trump "urged Turkey to deescalate, limit its military actions."

According to the statement, the U.S. President also "urged Turkey to exercise caution and to avoid any actions that might risk conflict between Turkish and American forces."

Turkish Minister of Foreign Affairs Mevlut Cavusoglu said Thursday that the White House statement did not reflect the true content of the phone conversation, it may have been prepared in advance.

"The U.S. message is more than clear, 'forget about Manbij and please wrap up Afrin!'" Faruk Logoglu, a former senior diplomat, told Xinhua.

"If the United States really means what it says, then there is risk of a serious rupture in Turkish-U.S. relations," he added, noting that much will depend on Turkey's response.

Top Turkish officials have repeatedly said that the army would also wipe out the YPG in Manbij as well as in the other two YPG-held cantons on the eastern part of the Euphrates River.

Turkish Prime Minister Binali Yildirim expressed Thursday Ankara's frustration with Washington's support for the Kurdish militia, regretting U.S. cooperation with terrorist organizations.

Underlining that is unacceptable for Ankara, he said Turkey would not allow any terrorist entity to exist along its border.

The United States has fought the Islamic State in Syria, along with the YPG as a reliable ground force, to whom it also provided arms.

Ties between the two NATO members have been highly strained for long due to U.S. military support to the Syrian Democratic Forces (SDF) of which the Kurdish militia constitutes the backbone.

Ankara sees the emergence of a YPG-controlled belt along its border with Syria as a major threat to national security.

Trump's homeland security adviser, Tom Bossert, was more blunt in his warning to Turkey as he spoke Thursday at the World Economic Forum in Davos in Switzerland.

Turkish troops' clash with the proxy forces the United States is using against IS would have a terrible outcome, especially if such a clash would take place in areas where U.S. advisors are present, Bossert was quoted as saying.

Any miscalculation and escalation could lead to grave consequences, he warned.

To Solmazturk, who chairs the Incek debates at the Ankara-based 21st Century Turkey Institute, the U.S. reaction to Turkey's operation is no surprise.

The United States is reacting to Ankara's cooperation with Moscow in Syria, where it is struggling for influence with Russia, he said.

Both analysts expected that international pressure over Ankara to end the operation should increase as days pass.

Cavusoglu said his U.S. counterpart Rex Tillerson had proposed to Ankara to establish a security zone of 30-km-deep along Turkey's border with Syria.

It is not possible to discuss such issues before trust is restored between the two NATO allies, Cavusoglu told the local media Thursday.

However, Tillerson denied later the same day that he had proposed a safe zone to Cavusoglu, maintaining that they had simply discussed a number of possible options.

Back in 2016, the United States had promised that the YPG forces in Manbij would withdraw to the eastern part of the Euphrates River.

The Turkish government previously said that the United States should take back the weapons it had already delivered to the YPG for any cooperation to be possible.

Erdogan said last month that Washington had provided a total of 4,900 truck-load of weapons to the Kurdish militia.

It is widely argued that the Turkish troops entered Afrin, where there were Russian troops in the past, following a deal with Russia, although Ankara dismissed the claim.

The Russian troops in Afrin left the area on Wednesday, local media said.

Logoglu felt that Trump, in a bid to maintain the U.S. influence, is making a strategic mistake in Syria by putting all his eggs in the YPG basket rather than working out an agreement with Turkey.