The Khurais oilfield produces about 1 percent of the world's oil, and Abqaiq is the company's largest facility - with the capacity to process 7 percent of the global supply.
Correspondents say they could have a significant impact on world oil prices.
"Such fruitless and blind accusations and remarks are incomprehensible and meaningless", foreign ministry spokesman Abbas Mousavi was quoted as saying in a statement.
Aramco said it would take 48 hours to give an accurate assessment of when production could be restored.
Traders and analysts said crude may spike to as high as $100 if Riyadh fails to quickly bring back supply.
The strikes sparked fires at the state-owned Aramco oil plants. The United States said it was also ready to tap emergency oil reserves if needed.
Oil price jump linked to Saudi Arabian attacks sparks energy stock rally
Brent, the worldwide oil benchmark, saw prices surge over 10% Monday to just shy of $67 a barrel. Pompeo said Tuesday that Trump was "prepared to meet with no preconditions".
The index .TASI opened down 2.3% but later pared some losses. This has caused 50% reduction in supply of ethane and other natural gases.
"Abqaiq is the nerve center of the Saudi energy system".
Saturday's attacks follow earlier cross-border attacks on Saudi oil installations and on oil tankers in Gulf waters.
Two crucial links to the world's oil supply were attacked by armed drones on September 14, with Yemen's Iran-allied Houthi rebels claiming responsibility.
There were no casualties in the attack but the full extent of the damage was not immediately clear as reporters were not allowed near the plants where Saudi authorities swiftly stepped up security. Iraq denied this on Sunday and vowed to punish anyone using Iraq as a launchpad for attacks. 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. Turkey, an ally of Iran, called for the avoidance of "provocative steps" that could harm regional stability.
Iran has stepped up its verbal attacks on the United States in reaction to President Donald Trump nixing the 2015 nuclear deal and ramping up economic sanctions against the Islamic Republic, which have hobbled its economy. Tehran ruled out talks until sanctions are lifted. "Amid all the calls for de-escalation, Iran has now launched an unprecedented attack on the world's energy supply", Pompeo said.
Saudi Arabia's Crown Prince Mohammad emphasized the country's "willingness and strength to thwart such a terrorist aggression and deal with its consequences", according to a statement carried by state news agencies.
The conflict has been in military stalemate for years. The war has become the world's worst humanitarian crisis.