The arrests happened over the past months, an Iranian intelligence official said at a news conference in Tehran. He said those taken into custody worked on "sensitive sites" in military and nuclear installations. The official did not say how many were given death sentences.
The announcement came amid weeks of rising tensions between Washington and Tehran over Trump's decision to pull the United States out of Iran's nuclear agreement with world powers last year and impose sweeping sanctions on the country.
The official said the 17 were recruited by the CIA and had "sophisticated training" but did not succeed in their sabotage missions. Their spying missions included collecting information at the facilities where they worked and installing monitoring devices, he said.
He said some were staff members at the targeted facilities, and the rest were working as consultant and contractors. The official said the CIA had promised them U.S. visas or jobs in America.
"That's totally a false story. That's another lie," Trump said at the White House.
Trump also said Iran has "disrespected" the United States, adding: "If they want to make a deal, frankly it's getting harder for me to want to make a deal with Iran because they've behaved very badly. They're saying bad things."
U.S. Secretary of State Mike Pompeo, a former CIA director, declined to address specifics of the arrests but said: "The Iranian regime has a long history of lying."
"I think everyone should take with a grain of salt everything that the Islamic Republic of Iran asserts today," he said.
With tensions rising recently, the U.S. has increased its military presence in the region and is sending at least 500 U.S. soldiers to Saudi Arabia, Iran's rival.
Last month, U.S. officials said American military cyberforces struck Iranian Revolutionary Guard computers, disabling systems that controlled its rocket and missile launchers. The cyberattack came after Trump backed away from an airstrike in response to Iran's downing of a U.S. surveillance drone.
The Iranian official did not give his name but was identified as the director of the counterespionage department of Iran's Intelligence Ministry. It is rare in Iran for intelligence officials to appear before media, or for any official to give a news conference without identifying himself.
The official said some of the agents recruited by the CIA had turned and are now working with his department against the United States.
He also handed out a CD with video of what Iran said was a foreign female spy working for the CIA. The disc also included the names of several U.S. Embassy staff members in Turkey, India, Zimbabwe and Austria who Iran said were in touch with the recruited Iranian spies.
Iran occasionally announces the detention of people it says are spying for foreign countries, including the U.S. and Israel. In June, Iran said it executed a former Defense Ministry employee convicted of spying for the CIA. In April, it said it uncovered 290 CIA spies inside and outside the country over the past several years.