The Syrian Democratic Forces (SDF) encircled Manbij on Friday, severing IS's principal supply route between Turkey and its de facto Syrian capital, Raqa.
The Syrian Observatory for Human Rights, a British-based monitor, said warplanes from a US-led coalition were conducting heavy bombing raids on Manbij.
"Tens of thousands of civilians who are still there can't leave as all the routes out of the city are cut," Observatory head Rami Abdel Rahman told AFP.
"ISIL terrorists now completely surrounded with no way out," he wrote on Twitter, using another name for IS.
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The city has been held by IS since 2014 and the Observatory said that thousands of inhabitants had recently fled Manbij as air strikes intensified.
"Bakeries in the town haven't been open since Friday and food is beginning to become rare," Abdel Rahman said.
He said that at least 159 IS fighters, 22 SDF troops and 37 civilians had been killed, mainly by bombing raids, since the alliance launched its Manbij offensive on May 31.