DOHA, Qatar—During President Barack Obama’s first term, some members of his National Security Council lobbied to pull a U.S. fighter squadron out of an air base in Qatar to protest the emirate’s support of militant groups in the Mideast.

The Pentagon pushed back, according to former U.S. officials involved in the discussion, saying a regional military command the U.S. maintains at the base was vital to American operations in the region. The issue was decided in late 2013 when the U.S. extended its lease on the base and didn’t...