Turkish troops head to the Syrian border on Aug. 27. (Halit Onur Sandal/AP)

Syrian rebels, backed by Turkey and the United States, pushed deeper toward U.S.-backed Kurdish positions in northern Syria on Monday, as Turkey’s foreign minister warned the Kurds to “immediately” pull back east of the Euphrates River or face more assaults.

The deepening animosities came a day after the Pentagon urged the American allies to stop fighting each other and refocus their efforts against the Islamic State, calling the clashes “unacceptable.”

But by Monday evening, the Syrian rebels said they had crossed the al-Sajour River, about nine miles north of the town of Manbij, which is controlled by Kurdish-aligned fighters. A Kurdish spokesman said the rebels had reached the river but had not crossed it.

Tensions have mounted since Turkish tanks, backed by U.S. and Turkish warplanes, pushed across the border into northern Syria last week, quickly seizing the Islamic State-controlled town of Jarabulus. In the following days, Syrian rebels of the U.S.-backed Free Syrian Army have swept southward and westward to target the Islamic State but have mostly clashed with Kurdish-aligned fighters.

On Monday, the Syrian Observatory for Human Rights, a British-based monitoring group, said areas south of Jarabulus were experiencing heavy bombardment by Turkish forces as well as clashes between the rebels and Kurdish-led groups. The group said that the rebels had seized 21 villages in the past four days and that the Turkish assaults had killed 40 civilians and wounded 70. Turkey has denied targeting civilians.

Turkey, which is battling an internal Kurdish insurgency, has long been concerned that the Syrian Kurds could try to forge an independent state along the border by linking two Kurdish enclaves in northwestern Syria. That, Ankara fears, could embolden Kurdish aspirations for self-determination in Turkey.

Washington has demanded that the People’s Protection Units (YPG), the military wing of the main Syrian Kurdish party, pull back to the east side of the Euphrates. The forces, which form the backbone of the U.S.-backed Syrian Democratic Forces (SDF), crossed the river and wrested control of Manbij, 25 miles south of Jarabulus, from the Islamic State this month.

YPG officials insist that their fighters have returned across the river. But Turkey and its proxies say they have not kept their promise.

“The YPG has to immediately cross east of the Euphrates River as they promised the United States and as they announced they would,” Turkish Foreign Minister Mevlut Cavusoglu said Monday. “If they don’t, they will be a target.”

The United States is now caught between the demands of two key allies. Turkey is a vital NATO ally, and the Kurdish fighters have played a critical role in rolling back the Islamic State in Syria’s five-year-old civil war. Washington’s biggest fear is that the animosities could divert resources and attention from fighting the Islamic State, also known as ISIS or ISIL.

“These clashes over the last couple of days are not helping us advance the efforts against Daesh,” John Kirby, a State Department spokesman, said Monday, using an Arabic acronym for the Islamic State.

Even as it supports Turkey’s demands for a Kurdish pullback, the United States has been careful not to alienate the SDF.

“We also fully support the SDF in their efforts to defeat ISIL,” said Peter Cook, a Pentagon spokesman. “The SDF have proven to be a reliable and capable force, and our support for the SDF in its fight against ISIL is ongoing and will continue to do so. They have fought hard and sacrificed to try and rid Syria of this hateful group.”

On Monday, the Jarabulus Military Council, a group created by the SDF, said its fighters would withdraw to areas south of the al-Sajour River — not east of the Euphrates, as Turkey wants.

“We declare the retreat of our forces to south, to al-Sajour River, to save civilian lives and so they [Turkey and Syrian rebels] have no other justification to target the villages and civilians,” the council said in a statement.

That could lead to more confrontations with Turkey and its Syrian proxies in and near Manbij.

Shervan Derwish, a spokesman for the local military council, which is aligned with the Kurds, denied reports that the YPG was sending reinforcements to the town to repel any attacks from Turkey or its proxies.

“We have our lines of defense, and we are ready to defend Manbij,” said Derwish, an ethnic Kurd who has also served as a spokesman for the SDF.

Abu Ibrahim, a Free Syrian Army commander, said he expects to fight the YPG. “We do not know whether the YPG is getting more reinforcements in Manbij, but what we know is that they didn’t retreat,” he said.

Carol Morello in Washington and Zakaria Zakaria in Gaziantep contributed to this report.