WASHINGTON—President Barack Obama secretly wrote Iran’s Supreme Leader Ayatollah Ali Khamenei in the middle of last month and described a shared interest in fighting Islamic State militants in Iraq and Syria, according to people briefed on the correspondence. Mr. Obama stressed to Mr. Khamenei that any cooperation on Islamic State was largely contingent on Iran reaching a comprehensive agreement with global powers on the future of Tehran’s nuclear program by a Nov. 24 diplomatic deadline, the same people say. The October letter marked at least the fourth time Mr. Obama has written Iran’s most powerful political and religious leader since...