The rebels' spokesman Yahya Saree said their operations "will expand and would be more painful as long as the Saudi regime continues its aggression and blockade" on Yemen, he said.The fires are now under control at both facilities, Saudi state media said.
A spokesperson for Iran's foreign ministry, Seyyed Abbas Mousavi, rejected the accusation that Tehran was behind the attacks, saying that "blind accusations and inappropriate comments in a diplomatic context are incomprehensible and meaningless".
Aramco is not only the world's biggest oil producer, it is also one of the world's most profitable businesses.
If the rebels were responsible for the attacks, their drones would have had to fly hundreds of miles from Yemen into central Saudi Arabia.
The strikes sparked fires at the state-owned Aramco oil plants.
The development comes as the United States has been a staunch supporter of Saudi Arabia's war campaign against civilians in Yemen since March 2015.
Later Saturday, however, U.S. Secretary of State Mike Pompeo blamed Iran for the attacks on the Saudi oil plants, and ruled out involvement by Yemen's Houthis. The US government is monitoring the situation and is committed to supplying stable and well oil to the global oil markets.
Crude oil futures shot up 9.5% to $60 as trading opened Sunday evening in NY, a dramatic increase.
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The spare capacity held by OPEC to supply consumers in the event of a significant production shortfall has been declining for decades as aging oil fields have lost production capacity.
The Yemeni forces on Saturday launched drone attacks on two plants at the heart of Saudi Arabia's oil industry, including the world's biggest petroleum processing facility. Their support "could be identified if we see any sharp recoveries in first 90 minutes of trading", he said.
It wasn't just Iran making threats.
The arch-foes were on the cusp of confrontation in June when Iran downed a U.S. drone and Trump ordered retaliatory strikes before cancelling them at the last minute. The Saudis attack the Houthis and the Houthis attack back.
Tensions between the US and Iran have escalated in recent months. since US President Donald Trump abandoned a deal limiting Iran's nuclear activities previous year and reinstated sanctions.
But Trump seemed to reject that idea Sunday night, tweeting: "The Fake News is saying that I am willing to meet with Iran, 'No Conditions.' That is an incorrect statement (as usual!)". While Trump has said he is ready for a meetup with no preconditions, US Treasury chief Steve Mnuchin said Thursday that such a meeting was not yet in the cards. "It's credible that the Iranians had something to do with this", said Maloney.
But in recent weeks, Trump has said he would be open to meeting with Rouhani, perhaps on the sidelines of the United National General Assembly in NY later this month.
However, he warned there could be a danger of Iran "overplaying" its hand.
Miller reported from Washington.