"Two thousand civilians evacuated Kobane tonight," Mustafa Bali, a spokesman for Kurds in Kobane, told AFP.
He said the residents, including women and children, had moved over the border to Turkey and he was himself now in a Turkish border village after the evacuation.
"(Kurdish) Military officials asked civilians to evacuate. They declared Kobane a military area," told AFP by telephone.
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"IS advanced to the eastern side. There were fierce clashes in the streets," he said.
Another Kurdish official in Kobane confirmed to AFP that street battles had started in the city.
"Street clashes started in Kobane. They (IS) are in the eastern suburb of Kobane," Idris Nahsen, deputy foreign minister of Kobane, told AFP by phone.
Turkish leaders have said that some 186,000 people, mainly Kurds, had already fled Kobane region over the last two weeks to Turkey to escape the advance of IS.