Iran's president and foreign minister also may skip next week's high-level meetings at the United Nations as the U.S. has yet to issue them visas, IRNA reported.
The U.N. meeting had been considered as an opportunity for direct talks between Iranian President Hassan Rouhani and President Donald Trump amid a summer of heightened tensions and attacks in the wake of America's unilateral withdraw from Iran's nuclear deal with world powers a year ago.
However, the recent attack in Saudi Arabia and hardening comments from Iran suggest such talks are increasingly unlikely.
Iran sent a note through Swiss diplomats in Tehran on Monday, reiterating that Tehran denies being involved in the Saudi attack, IRNA reported. The Swiss have looked after American interests in Tehran for decades.
"If any action takes place against Iran, the action will be faced by Iran's answer immediately," IRNA quoted the note as saying. It added that Iran's response wouldn't be limited to the source of the threat, without elaborating.
IRNA separately reported Wednesday that Iran's first delegation for the annual U.N. event had not left Iran due to not having visas. Foreign Minister Mohammad Javad Zarif was to travel to New York on Friday, with Rouhani following behind Monday, according to the agency.
As the host of the U.N.'s headquarters, the U.S. is mandated to offer world leaders and diplomats visas to attend meetings there. But as tensions have risen, the U.S. has put increasing restrictions on Iranians like Zarif.
The U.S. State Department did not immediately respond to a request for comment Wednesday.
U.S. Secretary of State Mike Pompeo is traveling to Saudi Arabia for meetings after Saturday's attack on a Saudi oil field and the world's largest crude oil processing plant. Saudi officials separately planned to share information about the weapons used in the attack they allege are Iranian.
Saudi Arabia also said on Wednesday that it joined a U.S.-led coalition to secure the Mideast's waterways amid threats from Iran after an attack targeting its crucial oil industry, while Rouhani told the kingdom it should see the attack as a warning to end its yearslong war in Yemen.
Yemen's Iranian-backed Houthi rebels have claimed the attack. The U.S. accuses Iran of being behind the assault, while Saudi Arabia already has said "Iranian weaponry" was used. Iran denies that.
"Almost certainly it's Iranian-backed," Prince Khalid bin Bandar, Saudi Arabia's ambassador to the United Kingdom, told the BBC. "We are trying not to react too quickly because the last thing we need is more conflict in the region."
The state-run Saudi Press Agency carried a statement Wednesday morning quoting an unnamed official saying the kingdom had joined the International Maritime Security Construct.
Australia, Bahrain and the United Kingdom already have joined the mission.
"The kingdom's accession to this international alliance comes in support of regional and international efforts to deter and counter threats to maritime navigation and global trade," the news agency said.
Cmdr. Joshua Frey, a spokesman for the U.S. Navy's 5th Fleet, declined to comment on the Saudi announcement, saying it "would be inappropriate to comment on the status of individual nations and the nature of any potential support."
The coalition aims to secure the broader Persian Gulf region. It includes surveillance of the Strait of Hormuz, the narrow mouth of the Persian Gulf through which a fifth of the world's oil travels, and the Bab el-Mandeb, another narrow strait that connects the Red Sea and the Gulf of Aden off Yemen and East Africa. Smaller patrol boats and other craft will be available for rapid response. The plan also allows for nations to escort their own ships through the region.
The U.S. blames Iran for the apparent limpet mine explosions on four vessels in May and another two in June sailing in the Gulf of Oman near the Strait of Hormuz, something Iran denies being behind. Iran also seized a British-flagged oil tanker and another based in the United Arab Emirates.
In Tehran, Rouhani told his Cabinet that Saudi Arabia should see the attack as a warning to end its war in Yemen, where it has fought the Houthi rebels since 2015 and sought to restore the internationally recognized government.
Rouhani said Yemenis "did not hit hospitals, they did not hit schools or the Sanaa bazaar," mentioning the Saudi-led coalition's widely criticized airstrikes.
He added that Iran does not want conflict in the region, but it was the Saudi-led coalition that "waged the war in the region and ruined Yemen."
"They attacked an industrial center to warn you. Learn the lesson from the warning," he said, portraying the Houthis as responsible for the drone strikes.
Wednesday's announcements comes after Saudi Arabia's energy minister said late Tuesday that more than half of the country's daily crude oil production that was knocked out by an attack had been recovered and that production capacity at its targeted plants would be fully restored by the end of the month.
Pompeo was due to land in the Red Sea city of Jiddah, where he was scheduled to meet with Saudi Crown Prince Mohammed bin Salman. Pompeo later will travel to the United Arab Emirates on Thursday to meet with Abu Dhabi's powerful crown prince, Sheikh Mohammed bin Zayed Al Nahyan. Both nations are U.S. allies and have been fighting against the Houthis in Yemen since March 2015.
The Saudi military planned to speak to journalists Wednesday in Riyadh to discuss the investigation into Saturday's attack "and present material evidence and Iranian weapons proving the Iranian regime's involvement." It did not elaborate.
Marine Gen. Joseph Dunford, chairman of the Joint Chiefs of Staff, said Tuesday that U.S. military experts were in Saudi Arabia working with their counterparts to "do the forensics on the attack" — gleaning evidence that could help build a convincing case for where the weapons originated.
On Wednesday, French President Emmanuel Macron's office announced experts from his nation would be traveling to Saudi Arabia to help the kingdom shed light " on the origin and methods" of the attacks. France has been trying to find a diplomatic solution to the tensions between Iran and the U.S., so any conclusion they draw could be used to show what a third-party assessed happened.