WASHINGTON — As fighters with the Islamic State bore down Tuesday on the Syrian town of Kobani on the Turkish border, President Obama’s plan to fight the militant group without being drawn deeper into the Syrian civil war was coming under acute strain.

While Turkish troops watched the fighting in Kobani through a chicken-wire fence, Turkey’s president, Recep Tayyip Erdogan, said that the town was about to fall and Kurdish fighters warned of an impending blood bath if they were not reinforced — fears the United States shares.

But Mr. Erdogan said Tuesday that Turkey would not get more deeply involved in the conflict with the Islamic State unless the United States agreed to give greater support to rebels trying to unseat the Syrian president, Bashar al-Assad. That has deepened tensions with President Obama, who would like Turkey to take stronger action against the Islamic State and to leave the fight against Mr. Assad out of it.

Mr. Erdogan has also resisted pleas to send his troops across the border in the absence of a no-fly zone to ward off the Syrian Air Force.