UNITED NATIONS—Iranian President Hassan Rouhani publicly proposed that the U.S. and Iran swap prisoners held in each other’s jails, including the Washington Post’s Tehran bureau chief Jason Rezaian, who Iran has charged with espionage.

The comments by the president, who made a string of public appearances at the United Nations in recent days, signaled a significant shift in his government’s position on Mr. Rezaian and other Americans detained in Tehran.

Iranian...