Saudi officials said the tankers were sabotaged.
The owner of the Norwegian vessel, Thome Ship Management, said the vessel had been "struck by an unknown object".
The crew were unharmed and the ship was not in any danger of sinking.
Saudi Energy Minister Khalid al-Falih, in comments run by state media, said the drone attack and Sunday's sabotage of four vessels, including two Saudi tankers, off Fujairah emirate, a major bunkering hub, threatened global oil supplies.
Picture taken June 19, 2018, in Abu Dhabi showing a drone which the Emirati armed forces say was used by Houthi rebels in Yemen in battles against the coalition forces led by the UAE and Saudi Arabia.
A loose connection between Iran and a series of drone "terrorism" strikes against pipeline infrastructure in Saudi Arabia caused oil prices on Tuesday to rise sharply: Brent climbed 1.7 percent at $71.39 per barrel, while West Texas Intermediate settled up 1.2 percent at $61.86 per barrel. Abizaid said starting a war over it would not be in the interest of Iran, the U.S., or Saudi Arabia.
Who was behind ship attacks? "It is easy to throw accusations but it is a hard situation, there are serious issues and among them is Iranian behaviour", he said, mentioning concern about Iran's missiles and regional policy. The UAE asked the United States to help investigate the damage, which Gulf officials have characterised as sabotage.
What has been the reaction?
Tuesday's attacks and Sunday's attacks on four tankers off Fujairah emirate have raised concerns that the United States and Iran might inching toward military conflict.
Pro-Iran Houthi militiamen, according to the statement, do not only pose a threat against Yemenis, but also undermine regional and worldwide security.
In Washington, Trump told reporters he would "absolutely" be willing to send troops to the Middle East, but that he's not planned for that and hopefully won't have to plan for that.
"I'm hearing little stories about Iran", Mr Trump said. "If they do anything, it would be a very bad mistake".
US President Donald Trump has warned Iran, saying that if Tehran does "anything" in the form of an attack "they will suffer greatly".
The UAE's foreign ministry said in its statement, published on WAM state news agency, that the concerned authorities have taken all the necessary measures, and are investigating the circumstances of the incident in cooperation with local and worldwide bodies.
The United Nations urged all sides to "exercise restraint for the sake of regional peace".
The incident comes amid fears in the Middle East that hawks in Washington are trying to provoke a military confrontation between the USA and Iran.
What is significance of location?
The incident happened 140 kilometres south of the Strait of Hormuz, where about a third of all oil traded by sea passes through.
The West Texas Intermediate for June delivery increased 0.74 USA dollar to settle at 61.78 dollars a barrel on the New York Mercantile Exchange, while Brent crude for July delivery rose 1.01 dollars to close at 71.24 dollars a barrel on the London ICE Futures Exchange.
Satellite images provided to the AP by Colorado-based Maxar Technologies showed a boom surrounding the Emirati oil tanker A. Michel, indicating the possibility of an oil leak.
Iranian officials have said that they could reach 20% enrichment within four days.
Trump wants to force Tehran to agree to a broader arms control accord and has sent an aircraft carrier and B-52 bombers to the Gulf in a show of force against what United States officials have said are threats to USA troops in the region. The Trump administration pulled America out of the deal past year.
US national security agencies said they believe proxies sympathetic to or working for Iran may have been responsible for the tanker attacks rather than Iranian forces themselves.