In another new initiative, the State Department is poised to expand its long-faltering campaign to counter the Islamic State’s propaganda machine, and one of the candidates being considered to lead the effort is Michael D. Lumpkin, a retired member of the Navy SEALs who is the Pentagon’s top Special Operations policy official.

The effort to overhaul the agency responsible for countering Islamic State messaging, the Center for Strategic Counterterrorism Communications, could draw on Mr. Lumpkin’s understanding of covert operations to improve the State Department’s efforts.

During the peak of the wars in Iraq and Afghanistan, nearly 13,000 Special Operations forces were deployed on missions across the globe, but a large majority were assigned to those two countries. Now, roughly half of the 7,500 elite troops overseas are posted outside the Middle East or South Asia, operating in 85 countries, according to the United States Special Operations Command.

There is other, subtler, evidence of the sway of senior Special Operations officers.

When Mr. Obama appeared before reporters in the Pentagon briefing room this month to discuss his administration’s strategy for fighting the Islamic State in Syria, he was flanked by a coterie of top national security officials, including Vice President Joseph R. Biden Jr., Defense Secretary Ashton B. Carter and Gen. Joseph F. Dunford Jr., the chairman of the Joint Chiefs of Staff.

Standing beside them was Gen. Joseph L. Votel, the head of the Special Operations Command, whose presence raised eyebrows at the Pentagon.

The threat from the Islamic State has become more prominent in the presidential campaign since the attacks in Paris and San Bernardino, Calif., and many candidates have proclaimed a need for more Special Operations troops to be deployed far and wide. Senator Marco Rubio, Republican of Florida, has talked about embedding Special Operations troops with Iraqi soldiers on the front lines, and Hillary Clinton said she would consider sending more special operators to Syria than the 50 that Mr. Obama recently authorized to assist rebels fighting Islamic State.

These calls for more American Special Operations troops have come even as some of the same candidates said they opposed boots on the ground in places such as Syria. Mr. Obama himself tried to draw a distinction during an interview this month with CBS News, when a reporter asked if recent Special Operations deployments in Iraq and Syria meant that he was reversing his pledge.