US Defence Secretary Mark Esper earlier announced the Pentagon was moving up to 1,000 troops away from the north after learning that Turkey was pushing further into Syria than previously expected.
Describing the situation there as "untenable", he said the SDF had been "looking to cut a deal" with the Syrian government and Russia to counter the Turkish attack.
This, he continued, would leave the US forces stuck between "two opposing advancing armies".
Hours after Mr Esper's comments, Syria said it was deploying its forces to the north to "confront a Turkish aggression". It is not yet clear where exactly the troops are being sent.
On Sunday, President Trump tweeted that it was "very smart" not to be involved in the fighting "for a change", saying engagement in Middle East conflict was a mistake.
Turkey is pushing deeper into northern Syria.
On Sunday, President Erdogan said his forces had already captured 109 sq km (42 square miles) of territory, including 21 villages.
He told reporters the key border town of Ras al-Ain had come under Turkish control - though the SDF said they had pushed Turkish forces back to the town's outskirts.
Mr Erdogan said Turkish forces had also besieged the town of Tal Abyad, some 120km (75 miles) away.
The UK-based monitoring group the Syrian Observatory for Human Rights (SOHR) said Turkey was in almost complete control there.
Both Ras al-Ain and Tal Abyad are key goals in the Turkish offensive against the Kurdish-led SDF forces.
Turkey also announced that its Syrian allies on the ground had seized a key motorway - called M4 - some 30-35km south of the border.
What are the casualty figures? They're rising, with civilians killed on both sides of the border:
More than 50 civilians and over 100 Kurdish fighters killed in north-eastern Syria, SOHR says SDF says the Kurdish forces' death toll is 56 and Turkey gives a higher figure of 440 Eighteen civilians killed in southern Turkey, according to Turkish reports Four Turkish soldiers and 16 pro-Turkish Syrian fighters killed in Syria, Turkey says
The fighting has spilled over to areas close to IS detainee camps.
Fears that Kurdish forces will be unable to keep IS prisoners confined appeared to have been realised when officials at the Ain Issa camp said nearly 800 relatives of foreign IS members had escaped.
The SOHR said the number of people who fled was 100. It not known where they have fled to.
The camp holds about 12,000 displaced people, previously including nearly 1,000 foreign women and children with jihadist links.
The SDF says it is currently holding more than 12,000 suspected IS members in seven prisons, and at least 4,000 of them are foreign nationals.
IS has claimed recent car bombings and on Saturday declared a new campaign in Syria.
Turkey says it will take responsibility for IS prisoners it finds during its offensive.