The fall of Kobane to the ISIS group would mark a major victory for the Terrorists, who are fighting for a long stretch of the border with Turkey for their self-proclaimed "Islamic caliphate".
At least 412 people, more than half of them from ISIS, have been killed in and around Kobane since mid-September, the Syrian Observatory for Human Rights said.
It warned that the inaccessibility of the area meant that the full toll was probably much higher.
With the fight for Kobane entering a crucial phase, the IS militants fought to extend their foothold into new areas in the south and west of the town, a day after piercing its Kurdish defense’s.
Gunfire, explosions and the roar of fighter jets were heard from the Turkish side of the border, while a Kurdish flag was seen flying in the centre of Kobane, according to an AFP journalist.
The IS Terrorists "are trying hard to capture the city", said Idris Nahsen, a Kurdish official still in Kobane, also known as Ain al-Arab.
He said they were meeting resistance from the Kurdish People's Protection Units.
"We need help from the international community," Nahsen told AFP by telephone. "Either we finish them (IS) or they will finish us."
IS fighters have seized a number of buildings in south and west Kobane, including a hospital under construction, according to the Observatory, a Britain-based monitoring group.
Kurdish fighters have ordered civilians to evacuate the town, after the Terrorists planted their black flags on its eastern side and entered Kobane on Monday.
In Iraq, meanwhile, at least 17 people were reported killed when a suicide bomber attacked an observation post used by Shiite militiamen watching for IS fighters crossing the Tigris River late Monday.
The United States and its allies have launched nearly 2,000 air raids against ISIS in both Iraq and Syria in an attempt to stop their advance, including four strikes in Kobane on Tuesday.
The Kurdish fighters are optimistic that their local knowledge of Kobane will compensate for their light weapons, said activist Mustafa Ebdi.
"They are fighting to defend their town and they say they will fight to the last person," he said.Ebdi added the latest US air raids had little effect.
"The strikes hit the Mishtenur area," he said, referring to a plateau south of Kobane."But they (IS) aren't gathered there. There are other places they should be hitting," he said.
The raging battle has prompted some 186,000 residents to flee across the Turkish border.
An official in the Turkish town of Suruc said on Tuesday that 700 people, including 47 wounded, had crossed the border from Syria overnight, both civilians and Kurdish combatants.
In northwestern Syria, meanwhile, a rival extremist organisation to IS -- the Al-Qaeda-affiliated Al-Nusra Front -- abducted a priest and several Christians on Sunday night, the Franciscan mission said on Tuesday.