Saudi Arabia, the Islamic republic's regional arch-rival, condemned "the acts of sabotage which targeted commercial and civilian vessels near the territorial waters of the United Arab Emirates", a foreign ministry source said.
Fujairah's port is located about 140km from the Strait of Hormuz, through which one-third of all oil at sea is traded.
Saudi Energy Minister Khalid al-Falih said in a statement that one of the two Saudi vessels attacked was on its way to be loaded with Saudi crude from Ras Tanura port for delivery to state-owned Saudi Aramco's customers in the United States.
Iran, embroiled in an escalating war of words with the United States over sanctions and the USA military's presence in the region, moved on Monday to distance itself.
Brent crude for July settlement rose $1.24 at $71.86 a barrel on the London-based ICE Futures Europe exchange at 1:58 p.m. local time.
The US Maritime Administration, which stressing the incidents had not been confirmed, warned shippers early Monday to exercise caution when travelling past Fujairah, a port city on the eastern edge of the United Arab Emirates on the Gulf of Oman.
UAE officials said earlier that an alleged sabotage attack targeted four boats.
It also comes as US Secretary of State Mike Pompeo scrapped a visit to Moscow and made a decision to travel to Brussels to discuss Iran with French, British and German officials.
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Saudi Arabia said two of its oil tankers were attacked while sailing toward the Persian Gulf, adding to regional tensions as the USA increases pressure on Iran. It said "the precise means of attack or sabotage is unknown". It called on the global community to assume its responsibilities to prevent such actions by parties attempting to undermine maritime traffic safety and security. It remains unknown if that number is still accurate or if it even includes the two ships Saudi Arabia have mentioned, although it likely does.
Falih said the attack aimed to undermine maritime freedom and the security of oil supplies to consumers all over the world.
Iran's Foreign Ministry spokesman Abbas Mousavi on Monday expressed "regret" for the sabotage of commercial vessels and asked for more details "on the exact extent of of this incident".
The Saudi Minister called on the worldwide community to protect maritime navigation and oil tanker security to prevent "adverse consequences" for energy markets and the global economy.
The tanker attacks are a signal of the growing political risk in the oil market since the USA ended waivers for crude purchases from Iran earlier this month.
Washington has also been increasing its military assets in the Middle East in the face of what it regards as new threats from Tehran.
Separately, in an early indicator of future output, USA energy companies last week reduced the number of oil rigs operating for the third time in four weeks.
"The oil market is reacting very sensitively to supply disruption risks considering the market is already tight", said Giovanni Staunovo, an analyst at UBS Group AG.