The threat comes as both Turkey and the Kurdish-led Syrian Democratic Forces (SDF) claim the other is violating terms of a 120-hour cease-fire brokered by Turkey and the U.S. on Thursday.
Violence had continued in northeast Syria despite the five-day peace agreement, a source told Fox News.
Dave Eubank with Free Burma Rangers, a private military company that provides emergency medical assistance, was on the ground near the Syrian border town of Ras al-Ayn trying to help trapped and wounded Kurds.
Eubank told Fox News the fighting hasn't stopped and movement in the area is severely limited, despite the cease-fire's intention to "pause" fighting to allow Syrian Kurds time and space to retreat from the area. Thousands of Kurdish civilians live in the so-called buffer zone, a senior military source had told Fox News.
The Turkish-backed Free Syrian Army (FSA) was "still shooting all through the night," Eubank said. "So far since [the] cease-fire, no airstrikes here, but artillery and ground attacks."
Erdogan threatened the Kurds on Saturday during a televised speech, saying they will be slaughtered if they don't pull back from the 20-mile-wide safe zone along the Turkey-Syria border by Tuesday night.
"We will start where we left off and continue to crush the terrorists' heads," Erdogan said.
A senior SDF official later said his forces will pull back from the border area and evacuate Ras al-Ayn, per the temporary peace agreement, as long as there are no delays.
Redur Khalil said forces will move back 19 miles from the border, withdrawing from the 75-mile area between Ras al-Ayn and Tal-Aybad.
Kurdish forces said they coordinated the withdrawal with the Americans. This is the first time the Kurds have publicly acknowledged they will pull back.
Khalil said a partial evacuation from Ras al-Ayn started Saturday after delays and coordinating with the U.S.
Turkey claimed on Saturday it was living up to the terms of the cease-fire agreement and accused the Kurds of violating it.
The Turkish Defense Ministry said Kurdish forces carried out 14 "provocative" attacks in Ras al-Ayn in 36 hours, according to the BBC.
In a statement, the SDF said there has been "no tangible progress" in solving the issues at the northeast border.
As of Friday, 86 civilians had been killed since Turkey launched its military offensive into Syria on Oct. 9, according to a war monitor, the BBC reported.
Erdogan claimed the move was to "neutralize terror threats" and establish a "safe zone." After carrying out airstrikes, Turkish ground troops later invaded northeastern Syria.
Nearly all U.S. troops there have been removed and will be redeployed in the region in the coming weeks.
The U.S. had teamed up with the Kurds to fight ISIS in the region. Some analysts and politicians criticized President Trump for removing America forces, saying it was a "green light" for Ankara to invade Syria and fight the Kurds.
Trump said the Turks have been "warring for many years," and that the U.S. does not need to protect war-torn Syria because it's "7,000 miles away."
The president on Friday claimed "thousands and thousands" of lives were being saved in Syria and Turkey due to the cease-fire.
Fox News' Griff Jenkins and Jennifer Griffin contributed to this report, as well as The Associated Press.