The assault began on Saturday after nearly 20,000 people were evacuated from the ISIS-held area in the eastern province of Deir el-Zour along the Iraqi border, according to SDF spokesman Mustafa Bali.
"After 10 days of evacuating more than 20,000 civilians ... the battle was launched tonight" to wipe out the last remnants of the organization, it said.
"We do not think he is in Syria", Bali said, without adding further details about the whereabouts of the man who declared a cross-border IS "caliphate" in 2014. USA -led coalition warplanes are giving cover to advancing SDF fighters.
The U.S. military presence in Syria has offered the Kurds a measure of protection against any Turkish offensive.
Mustafa Bali, the head of the SDF media office, told Reuters the aim of the assault was to "eliminate the last remnants of the terrorist organisation" and called it the "last battle".
In December, Trump declared victory over the Islamic State terror group in war-torn Syria and announced that American forces would be "coming home soon", sparking concern among officials in Washington and allies overseas.
The SDF, backed by a US -led coalition, began the assault on Saturday, seeking to wipe out the last remnants of the jihadist group's "caliphate" in the SDF's area of operations in eastern and northern Syria. "This was the foreign fighter element, some of his most trusted people", the intelligence source reportedly said.
After pausing more than a week to allow tens of thousands of civilians to flee the town, the SDF on Saturday renewed its push to wrest the last 4 square kilometers (1.5 square miles) from the militants.
In December, US President Donald Trump said IS militants were "mostly gone" and announced the US would withdraw all of its 2,000 troops from Syria.
Mr Trump said last December that he had ordered a complete troop withdrawal from Syria, a shock announcement that left the Kurds scrambling for new allies.
"These are ideal figures for [IS] to use in their negotiation for a safe corridor to escape", the official said.
The SDF units claim that Baghuz is the last village held by the terror group in Syria.
US and SDF officials say, however, they have no reason to believe he is there.
At least 48 suspected ISIS members were among them, according to the Britain-based Syrian Observatory for Human Rights.
While the Kurdish People's Protection Units (YPG) have been a key United States ally in the fight against IS, Ankara views them as "terrorists".
SDF-held areas make up a third of the whole country, and Damascus has repeatedly said it would eventually see them revert to government control.
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