A senior US administration official confirmed that "our detained US citizens have been released and that those who wished to depart Iran have left."
The Iranian report said those on board the "special Swiss plane" -- Jason Rezaian, the Washington Post's Tehran correspondent; Saeed Abedini, a Christian pastor; former US Marine Amir Hekmati; and Nosratollah Khosravi -- departed "on a special Swiss plane."
The US official did not say who was aboard the plane, but the Washington Post said that those leaving included Rezaian, who has been held for nearly 18 months.
The Post reported that the flight out of Iran was delayed because Rezaian's mother Mary and wife Yeganeh, who also were on the airplane, initially did not appear on the flight manifest.
US Secretary of State John Kerry told reporters traveling with him from Vienna to Washington that Iranian Foreign Minister Javad Zarif resolved the problem.
"Zarif had no question about it," Kerry was quoted as saying. "It was part of the agreement, it was clearly stated. The problem was, one of the guys on the ground, at a military base, didn't have it on the manifest."
Iran announced their release on Saturday, just hours before Tehran's historic nuclear deal with world powers was implemented, in exchange for Washington pardoning seven Iranians accused of sanctions-busting.
State television said the seven Iranians -- Nader Modanlou, Baharam Mechanic, Khosrow Afghahi, Arash Ghahreman, Tooraj Faridi, Nima Golestaneh, and Ali Saboonchi -- "will be freed today."
A fifth American was also released in a separate process, a US official has said.
Amir Hekmati, a former US Marine who faced a death sentence as an alleged spy, also was released, his family said, adding that they were officially told he was on a plane leaving Iran.
"It is hard to put into words what our family feels right now. But we remain in hopeful anticipation until Amir is in our arms."
Washington Post publisher Frederick Ryan said in a statement: "We are relieved that this 545-day nightmare for Jason and his family is finally over."
The 39-year-old Rezaian, a dual US-Iranian citizen born in California, was detained in Iran on July 22nd, 2014.
The Post statement said the paper was "enormously grateful to all who played a role" in securing Rezaian's release, and the paper thanked those "around the world who have spoken out on Jason's behalf and against the harsh confinement that was so wrongly imposed upon him."
Rezaian "will be reunited with his family, including his brother Ali, his most effective and tireless advocate," the Post said.
Iran's ambassador to the United Nations, Gholam Ali Khoshroo, said on Saturday that Switzerland played a "positive role" in the prisoner swap.
The UN nuclear watchdog announced late Saturday that Iran had complied with its side of the July 2015 accord, allowing the lifting of sanctions.