The oil processing plant at Abqaiq handles crude from the world's largest conventional oilfield, the supergiant Ghawar, and for export to terminals Ras Tanura - the world's biggest offshore oil loading facility - and Juaymah.
Trump further said that the U.S. was awaiting confirmation from Saudi Arabia on who they suspect to be behind the attack on oil facilities, following which the further course of action would be determined.
President Trump announced on Sunday that the United States is "locked and loaded" and ready to respond to the weekend attack on Saudi oil refineries, but is waiting on the Saudis to determine who is to blame.
Oil prices surged to six-month highs on Monday while Wall Street futures fell and safe-haven bets returned after Sunday's attacks. USA crude futures jumped as much as 15% before hovering around 11%.
Brent futures, the worldwide benchmark, jumped by almost 20% to US$71.95 a barrel, while West Texas Intermediate increased by 15% to $63.34 before falling back in trading.
Conway said the attack on the Saudi oil fields "did not help" the prospects for Trump talks with Rouhani, but "he'll consider it". The kingdom produces approximately 10% of the total global supply of 100 million barrels per day. Iran wields significant influence in southern Iraq, which is situated much closer than Yemen to the affected Saudi sites.
Still, even if the plant goes back online and there is no fundamental change to the world's supply of oil, prices may move higher and stay higher because traders would build in a "security premium", said Michael Lynch, president of Strategic Energy & Economic Research. "Saudi Arabia's oil fields have been the target of a number of drone attacks over the past year, with Yemen's Houthi rebels claiming responsibility for the latest attack".
As markets opened Monday in Asia, the Singapore Exchange saw Brent crude trading at more than 71 US dollars per barrel on the oil futures markets, up 19 percent from Friday's close.
The attack, which knocked out more than half of the Saudi oil output, may force the U.S.to tap into its own oil reserves to keep the markets well supplied.
During the weekend, the International Energy Agency stressed that global markets would not be disrupted by the Saudi attacks.
"Abqaiq is perhaps the most critical facility in the world for oil supply". Yemen's Houthi rebels are claiming responsibility for the attack. They also increased overall tensions in the region amid an escalating crisis between the USA and Iran over Tehran's unraveling nuclear deal with world powers.
The Saudi interior ministry had confirmed that the attacks caused fires at the two facilities. Initial reports indicated the attack came from Yemen.