- Video report by ITV News Correspondent Emma Murphy
Speaking at the White House, President Trump said the US is not looking at retaliatory options until he has "definitive proof" that Iran was responsible.
He went on to tell reporters the US "is prepared” if the attacks warrant a response. It comes after he said the nation's military is "locked and loaded" to respond to the attack against its Middle East ally.
Secretary of state Mike Pompeo also said "emerging information indicates that responsibility lies with Iran".
Iran has denied involvement, though it comes amid heightened tensions over Tehran's unravelling nuclear deal with world powers, including the US, which pulled out of the deal last year.
Mr Trump said Mr Pompeo will be travelling to Saudi Arabia but did not say when.
American officials released satellite images of the damage at the heart of the kingdom's crucial Abqaiq oil processing plant and a key oil field, and US officials said the attacker used multiple cruise missiles and drone aircraft.
The Americans alleged the pattern of destruction suggested Saturday's attack did not come from neighbouring Yemen, as claimed by Iranian-backed Houthi rebels there.
A Saudi military spokesperson later made a similar accusation, alleging "Iranian weapons" had been used in the assault.
Iran rejected the allegations, and a government spokesperson said there was now "absolutely no chance" of a hoped-for meeting between Iranian President Hassan Rouhani and Mr Trump at the UN General Assembly next week.
For his part, Mr Trump sent mixed signals, saying his government was "locked and loaded" as it waited for Saudi confirmation of Iran being behind the attack, then later tweeting that the US did not need Middle Eastern oil "but will help our Allies!"
The new violence has led to fears that further action on any side could rapidly escalate a confrontation that has been raging just below the surface in the wider Persian Gulf in recent months.
There already have been mysterious attacks on oil tankers that Washington blames on Tehran, at least one suspected Israeli strike on Shiite forces in Iraq, and the downing of a US military surveillance drone by Iran.
Those tensions have increased since Mr Trump pulled the US out of Iran's 2015 agreement with world powers that curtailed Tehran's nuclear activities and the US reimposed sanctions that sent Iran's economy into freefall.
Benchmark Brent crude prices gained nearly 20% in the first moments of trading Monday before settling down.
The attack halted production of 5.7 million barrels of crude a day, more than half of Saudi Arabia's global daily exports and more than 5% of the world's daily production.
Yemen's Houthi rebels, who have been targeted by a Saudi-led coalition since March 2015 in a vicious war in the Arab world's poorest country, maintain they launched 10 drones that caused the extensive damage.
They reiterated that Saudi oil sites remain as targets, warning foreign workers to stay away.
US officials say the damage to the north-facing parts of the facilities suggest the attack came across the Persian Gulf from Iraq or Iran.