The verdict and the sentence are not yet known.
Rezaian's family said the lack of information "follows an unconscionable pattern by Iranian authorities of silence, obfuscation, delay and a total lack of adherence to international law, as well as Iranian law."
ISNA quoted the judiciary spokesman, Gholam-Hossein Mohseni Ejei, as saying that "this person has been sentenced, but I don't know the details of the verdict."
Mohseni Ejei said the ruling may be appealed by Rezaian or his lawyer in the next 20 days.
Rezaian, The Washington Post's bureau chief in Tehran, was detained in Iran in July 2014 and has languished in jail for over a year despite an international outcry.
Uncertainty continued to reign on Sunday. The statement from his family called it "another sad chapter in his 14-month illegal imprisonment and opaque trial process."
And Doug Jehl, the foreign editor of the Post, said on CNN's "Reliable Sources" that the mysterious announcement shows that "what we're seeing unfolding here is sham."
"For Iran to say that there's been a verdict but it's not final simply suggests, again, that this is not a matter for the courts, it's a matter that's being decided in the political spheres in Iran," he said.
John Kirby, a spokesman for the U.S. State Department, said the U.S. government has no firm information on the apparent ruling.
"We're monitoring the situation closely, and we continue to call for all charges against Jason to be dropped and for him to be immediately released," Kirby said.
Rezaian's case has become a symbol of the increased dangers faced by journalists around the world. Rezaian, who has dual Iranian and American citizenship, has been detained in Iran longer than any American journalist in the past.
Marty Baron, the paper's top editor, said it is unclear whether the apparent sentence has even been shared with Rezaian or his lawyer.
"This vague and puzzling statement by the government of Iran only adds to the injustice that has surrounded Jason's case since his arrest 15 months ago," Baron said in a statement.
Rezaian's ordeal began last July when he and his wife, Yeganeh Salehi, were taken into custody. She was later released.
For months Rezaian was not told of the charges against him. Prosecutors eventually accused him of espionage and other offenses, including "collaborating with a hostile government" and "propaganda against the establishment," according to the Post.
The newspaper steadfastly denied the allegations, calling them "the product of fertile and twisted imaginations." And the State Department called the charges "absurd."
The trial reportedly concluded back in August with no announcement of a ruling.
At times Rezaian has seemed like a pawn in a geopolitical faceoff. U.S. President Barack Obama faced criticism from some for concluding a deal between major world powers and Iran over Iran's nuclear program without any pledge from Iran that it would release Rezaian and other Americans held in Iran.
In a speech this summer, Obama mentioned Rezaian and other Americans "who are unjustly detained in Iran."
"Journalist Jason Rezaian should be released," the President said.
Boxing great Muhammad Ali, an American Muslim, also urged Tehran to free Rezaian on bail.
"To my knowledge, Jason is a man of peace and great faith, a man whose dedication and respect for the Iranian people is evident in his work," Ali said in a religiously worded statement issued in March.
Iran's human rights chief, Mohammad Javad Larijani, told news outlet France 24 last year that he hoped Rezaian's case would come to a positive conclusion. "Let us hope that this fiasco will end on good terms," he said.
The Post has sought to keep world attention focused on the case. On Friday, it noted that Rezaian had been detained for 444 days -- "the same amount of time as U.S. government employees during the Iran hostage crisis of 1979-1981." The Post called this "a milestone significant in its injustice."
Jehl said on Sunday that the lack of a publicized ruling "suggests once again that Jason is not really a prisoner, he's a bargaining chip being used by the Iranian government to extract some concessions from the U.S."