IS fighters seized part of a strategic hill overlooking the town late yesterday, a monitor said, but their progress was slowed by new strikes from the coalition of Washington and Arab allies.
The dusty town on the Turkish border has become a crucial battleground in the international fight against IS, which sparked further outrage this weekend with the release of a video showing the beheading of Briton Alan Henning.
The video -- the latest in a series of on-camera beheadings of Western hostages -- included a threat to another hostage, US aid worker Peter Kassig, whose parents made an impassioned plea for his release.
"IS succeeded on Saturday night in taking the southern part of the Mishtenur hill," Observatory director Rami Abdel Rahman told AFP.
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If the jihadists seize the hilltop, he said, "the whole town of Kobane will be in their sights and it will be easier to take."
Abdel Rahman said seven new coalition strikes against IS positions were carried out in the area late yesterday and the strikes were hindering the jihadists' advance.
Dozens of fighters from IS have been reported killed in the latest coalition raids.
The Observatory, which relies on a network of local sources, said at least 33 IS fighters and 23 of the town's Kurdish defenders were killed yesterday.
IS began its advance on Kobane on September 16, seeking to cement its grip over a long stretch of the Syria-Turkey border.
The offensive has prompted a mass exodus of residents from the town and the surrounding countryside, with some 186,000 fleeing into Turkey.