About 50 per cent of the production cut by the attack was restored by Tuesday.
An Iran-aligned rebel group known as the Houthi movement has claimed responsibility for the attack.
Iran has denied any involvement, calling Trump’s claim “maximum lies.”
Trump originally suggested the U.S. would initiate a military response but stopped short of any legitimate retaliation.
The dispute may have effectively ended speculation about a possible U.S.-Iran meeting at the United Nations later this month. The meeting would be the first positive step since relations deteriorated over economic and nuclear disagreements.
Trump has since ordered his administration to “substantially increase sanctions” on Iran.
Iran has warned that any action taken against it by the U.S. will “immediately” be met with their response.
Drone attacks were launched on stated-owned Saudi Aramco oil facilities Abqaiq and Khurais on Sept. 14. Abqaiq is considered the world’s largest oil processing plant.
The co-ordinated strikes sparked massive fires at the sites and sent thick plumes of smoke into the air.
WATCH: Trump says U.S. knows who was behind recent Saudi oil attacks
While no one was injured, the attacks took out nearly half of Saudi Arabia’s oil production, effectively putting five per cent of the global daily output on hold.
Officials initially believed the kingdom’s oil operations could be restored within a few days, but hope for that has dwindled as the damage assessment continued. While 50 per cent of operations resumed Tuesday, the remaining portion likely won’t be repaired until the end of September.
Houthi rebels say they launched the strikes in retaliation to Saudi Arabia’s military effort against them in Yemen. The group has been locked in a war against a Saudi-led coalition that has fought to reinstate the Yemeni government since 2015.
This isn’t the first time Houthi rebels have used drones in combat since the start of the Saudi-led war. More recently, the Houthis launched drone attacks targeting Saudi Arabia’s crucial East-West Pipeline in May. In August, Houthi drones struck Saudi Arabia’s Shaybah oilfield.
Since the weekend assaults, the group has threatened to carry out more attacks and claimed its weapons could reach anywhere in Saudi Arabia.
Why does it matter?
The strikes, which temporarily halved Saudi Arabia’s oil production, are considered the largest single supply disruption in half a century.
The attack has already had a significant impact on the global oil industry — prices surged as much as 20 per cent at one point on Monday.
With fears that full production could be offline for two to three weeks, analysts believe the impacts could widen and intensify.
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“Saudi Arabia is the only country in the world who has spare capacity for crude,” Roger McKnight, a senior petroleum analyst at En-Pro International, told Global News.
“When you knock out five per cent of the country’s capability, you really knocked out all the spare capacity of crude for the global community.”
The U.S. has suggested it would consider offsetting the effects by drawing on the country’s emergency oil reserve.
In tweets, Trump said he would draw on the reserve “if needed.”
That may not be enough, said McKnight.
“The crude oil inventories in the U.S. are down two per cent versus the five-year average. This is going to slow things down again,” he said.
McKnight and other industry analysts in Canada anticipate a five- to six-cent hike in gasoline prices by Wednesday.
But the long-term impacts for Canadian drivers — and the global oil market — rest on how long it takes the Saudis to restore full operations.
“It’s really serious,” McKnight said. “We don’t know how long this is going to be going on.”
Why is the U.S. blaming Iran?
The U.S. claims the pattern of destruction in the attack did not come from neighbouring Yemen, as claimed by Houthi rebels.
A Saudi military official has also alleged “Iranian weapons” were used.
Trump hinted at a military response to the attacks one day after the drone strikes.
He tweeted: “There is reason to believe that we know the culprit, are locked and loaded depending on verification, but are waiting to hear from the Kingdom as to who they believe was the cause of this attack.”
White House officials later downplayed the threat of action, saying the president’s language was merely “a reflection” of his desire to protect the U.S. “from these sorts of oil shots.”
One day later, the U.S. president softened his response, stressing that he did not want to go to war and there was “no rush” to do so.
“We have a lot of options but I’m not looking at options right now,” he said. “We want to find definitively who did this.”
He later doubled down on his claim Iran was behind the attacks.
“Well, it’s looking that way,” he told reporters at the White House. “That’s being checked out right now.”
What does Iran have to say?
Iranian President Hassan Rouhani has claimed the strikes were carried out by “Yemeni people” retaliating for attacks in the Saudi-led war.
He told Reuters reporters: “Yemeni people are exercising their legitimate right of defence.”
The country’s foreign minister, Javad Zarif, called the accusations stemming from the U.S. “max deceit.”
Iran’s supreme leader, Ayatollah Ali Khamenei, ruled out talks with Washington on Tuesday.
“There will be no talks with the U.S. at any level,” Khamenei was quoted as saying by Iranian state TV.
The latest breakdown steers the two countries away from attempting to quell tensions, which were renewed by economic sanctions and Trump’s withdrawal from the 2015 Iran nuclear deal.
While the talks may be on hold for now, Khamenei said that if the U.S. returns to the deal, Iran would reconsider negotiations.
“Otherwise, no talks will happen … with the Americans,” he said. “Neither in New York nor anywhere.”
— With files from the Associated Press and Reuters