The AFP news agency quoted a spokesman for Kurdish forces as saying that all civilians had been told to leave immediately.
"Military officials asked civilians to evacuate - they declared Kobane a military area," said the spokesman identified by AFP as Mustafa Bali. "ISIL advanced to the eastern side. There were fierce clashes in the streets," he added.
Also, Al Jazeera reported that the clashes in Qani Arab are the heaviest since the start of the offensive as ISIL tries to bring in more fighters to take control of the city.
Kurdish activists stated that one group of ISIL fighters were eliminated by the town's defenders, drawn from the Kurdish irregular force known as the YPG, as they attempted to enter from the east.
The advances came after ISIL fighters seized part of Mishtenur Hill, which overlooks Kobane, late on Saturday, although US-led air attacks tried to slow the fighters’ advances.
According to Reuters, at least 30 Kurdish fighters were killed in two suicide attacks on two checkpoints run by Kurds in Syria's northeastern city of Hasakah.
Kobane has become a crucial battleground in the international fight against ISIL fighters. The battle has forced an estimated 186,000 civilians to flee to Turkey.