Outgunned Kurdish militia were struggling to prevent the terrorists closing off the last escape route for civilians still in the area, prompting an appeal for urgent military assistance.
US-led warplanes have intensified air strikes against IS, which has been attacking Kobane for three weeks, but the Pentagon has warned that, without a ground force to work with, there are limits to what can be done.
Neighbouring Turkey has so far refrained from any action against the ISIS on its doorstep, despite four straight nights of protests among its own large Kurdish minority that have left 31 people dead.
The ISIS advance has brought the front line to just 1.3 kilometres from the border.IS now controls 40 percent of Kobane, according to the Britain-based Syrian Observatory for Human Rights.
"The capture of the headquarters will allow the ISIS to advance on the border post with Turkey to the north of the town," its director Rami Abdel Rahman told AFP.
"If they achieve that, they will have the Kurdish forces inside Kobane completely surrounded."
As fighting raged, an IS militant carried out a suicide car bomb attack to the west of the Kurdish headquarters that killed two defenders, and 10 Kurds were killed in an IS ambush on the south side of town, the Observatory said.
US planes conducted nine new airstrikes in Syria on Thursday and Friday, the US military said.And the Observatory said four strikes hit in the vicinity of the Kurdish headquarters on Friday afternoon.
Thousands could be 'massacred'
UN envoy for Syria Staffan de Mistura warned that 12,000 or so civilians still in or near Kobane, including 700 mainly elderly people in the town centre, "will most likely be massacred" by IS if the town falls.
Kobane was "literally surrounded," except for one narrow entry and exit point to the border, de Mistura said.
United Nations special envoy for Syria Staffan de Mistura urged Turkey to allow Kurds to cross into Syria to protect Kobane against ISIS fighters.
The envoy called on Turkey, "if they can, to support the deterrent actions of the coalition through whatever means from their own territory."
"We would like to appeal to the Turkish authorities in order to allow the flow of volunteers at least, and their equipment to be able to enter the city to contribute to a self-defence operation," he said in Geneva.
The statement marked an unusual one by the United Nations, which usually strives to stay neutral in conflicts, but de Mistura explained the rare appeal by the precarious situation in the border town.
The coordinator of the US-led campaign against IS, retired US general John Allen, was to hold a second day of talks in Ankara after the NATO ally insisted it could not be expected to take ground action against the ISIS alone.
'Willing to die'
Kobane activist Mustafa Ebdi said the IS militants were using civilian cars with Kurdish flags to avoid coming under attack by coalition aircraft.
He said the risk of Kobane falling was high, and described the town as a "symbol of resistance to IS in Syria, every Kurdish fighter is willing to die," he said.
With the world media gathered just across the border, the conquest of Kobane would be a highly visible symbolic victory for the extremists.
Ankara has been deeply reluctant to allow weapons or Kurdish fighters to cross the border.
Salih Muslim, leader of the main Kurdish political party in Syria, the Democratic Union Party, told AFP the Kurds are not a threat to Turkey and called for urgent help.
"It would be very good if Turkey urgently opens its soil for the passage of military weapons, not Turkish soldiers, for Kobane," he said.