The Obama administration is pressing Turkey to deploy thousands of additional troops along its border with Syria to cordon off a 60-mile stretch of frontier that U.S. officials say is used by Islamic State to move foreign fighters in and out of the war zone.

The U.S. hasn’t officially requested a specific number of soldiers. Pentagon officials estimated that it could take as many as 30,000 to seal the border on the Turkish side for a broader humanitarian mission. Cordoning off just one section alone could take 10,000 or more,...