WASHINGTON—Iran’s paramount political figure, Supreme Leader Ayatollah Ali Khamenei, has responded to overtures from President Barack Obama seeking better relations by sending secret communications of his own to the White House.

The Iranian cleric wrote to Mr. Obama in recent weeks in response to an October presidential letter that raised the possibility of U.S.-Iranian cooperation in fighting Islamic State if a nuclear deal is secured, according to an Iranian diplomat. The supreme leader’s response was “respectful” but noncommittal,...