One of the two vessels was on its way to be loaded with Saudi crude oil from the port of Ras Tanura, to be delivered to Saudi Aramco customers in the US.
The UAE foreign ministry has said there were no casualties and the Fujairah port operations were normal.
- The UAE and Saudi Arabia have sought to find other routes to bypass the Strait, including building more oil pipelines.
He emphasized the joint responsibility of the worldwide community to protect the safety of maritime navigation and the security of oil tankers, as these incidents pose a danger to energy markets and the global economy.
One of the ships was holed at the waterline.
"This criminal act constitutes a serious threat to the security and safety of maritime navigation and adversely impacts regional and global peace and security", the source added.
A spokesman for the Iranian Foreign Ministry also warned about a "conspiracy orchestrated by ill-wishers" and "adventurism by foreigners" meant to undermine regional stability. Oil prices rose on world markets on Monday with benchmark Brent North Sea crude up 1.8 per cent at $71.90 a barrel in London.
Saudi Arabia's foreign ministry in a separate statement voiced support for the UAE, the Middle East's trade and business hub.
Mr Mousavi was cited by the semi-official ISNA news agency as saying "such incidents have negative impact on maritime transportation security", and asking for regional countries to be "vigilant against destabilising plots of foreign agents".
The US issued a new warning to sailors as the UAE's regional allies condemned Sunday's alleged attack that the UAE says targeted four ships off the coast of its port city of Fujairah.
The U.S. has warned that "Iran or its proxies" may be targeting maritime traffic in the region.
Falih said the attack aimed to undermine maritime freedom and "security of oil supplies to consumers all over the world".
The United States reimposed sanctions in November on exports of Iranian oil after US President Donald Trump, last spring, unilaterally pulled out of the 2015 accord between Iran and six world powers to curb Tehran's nuclear programme. India and seven other countries who had got a sanctions waiver from the United States for the past six months were refused an extension of the waiver - this meant India had to stop buying Iranian oil including closing down the rupee account that it used to pay for Iranian oil. The US also refused to allow Iran to export heavy water surpluses to Oman or swap low-enriched uranium that exceeded 300 kg for yellowcake uranium.
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