WASHINGTON—Hillary Clinton, in her last months as secretary of state, helped open the door to a dramatic shift in U.S. policy toward Iran: an acceptance that Tehran would maintain at least some capacity to produce nuclear fuel, according to current and former U.S. officials. In July 2012, Mrs. Clinton’s closest foreign-policy aide, Jake Sullivan, met in secret with Iranian diplomats in Oman, but made no progress in ending the dispute over Iran’s nuclear program. In a string of high-level meetings here over the next six months, the secretary of state and White House concluded that they might have to let...