ISTANBUL — Islamic State militants pushed on Monday into the eastern edge of the Syrian Kurdish town of Kobani on the Turkish border, after sustained shelling that drove back the Kurdish fighters and Syrian insurgents fighting alongside them, killing 16 and raising fears of a massacre of civilians, Kurdish fighters and activists said.

Anwar Muslim, a coordinator in Kobani for the People’s Protection Committees, a Kurdish militant group known as the Y.P.G., said Monday night that 12,000 civilians were trapped inside the town. He said that his group was running out of heavy ammunition, and with Islamic State militants close by the population was in constant fear of car bombs or suicide bombers.

Rooz Bahjat, who identified himself as a senior security official with the Kurdistan Regional Government in Iraq, said that as many as 9,000 Islamic State fighters were closing in on the area, leaving other fronts in eastern and northern Syria and even as far away as Iraq to attack what he called “a bastion of democracy and secularism” in Kobani, which has given shelter to internally displaced Syrians from a wide range of ethnic groups.

“The whole might of ISIS right now is turned onto Kobani,” he said.

His information could not be independently verified, but Kurdish officials have been warning for days that without more military aid, Kobani will fall. And Islamic State fighters in Raqqa and Aleppo Provinces have said in recent interviews that they were redeploying toward Kobani after having faced United States-led airstrikes in their strongholds elsewhere in Syria.