Iran has rejected US charges it was behind the strikes on Saturday that damaged the world's biggest crude-processing plant and triggered the largest jump in crude prices in decades.
Relations between the United States and Iran have deteriorated since Trump pulled out of the Iran nuclear accord last year and reimposed sanctions over Tehran's nuclear and ballistic programmes. Washington also wants to pressure Tehran to end its support of regional proxy forces, including in Yemen where Saudi forces have been fighting Iran-backed Houthis for four years.
The United States was still investigating if Iran was behind the Saudi strikes, Trump said, but "it's certainly looking that way at this moment."
Trump, who has spent much of his presidency trying to disentangle the United States from wars he inherited, made clear, however, he was not going to rush into a new conflict on behalf of Saudi Arabia.
"I'm somebody that would like not to have war," Trump said.
Several US Cabinet members, including Secretary of State Mike Pompeo and Energy Secretary Rick Perry, have blamed Tehran for the strikes. Pompeo and others will travel to Saudi Arabia soon, Trump said.
A day after saying the United States was "locked and loaded" to respond to the incident, Trump said on Monday there was "no rush" to do so.
"We have a lot of options but I'm not looking at options right now. We want to find definitively who did this," he said.
Iranian President Hassan Rouhani said the strikes were carried out by "Yemeni people" retaliating for attacks by a Saudi-led military coalition in a war with the Houthi movement.
"Yemeni people are exercising their legitimate right of defence," Rouhani told reporters during a visit to Ankara.
Iranian Foreign Ministry spokesman Abbas Mousavi called the allegations "unacceptable and entirely baseless."
The attacks cut 5% of world crude oil production.
Oil prices surged by as much as 19% after the incidents but later came off their peaks. The intraday jump was the biggest since the 1990-91 Gulf crisis over Iraq's invasion of Kuwait.
The market eased from its peak after Trump said he would release US emergency supplies and producers said there were enough stocks stored up worldwide to make up for the shortfall. Prices were around 12% higher by afternoon in the United States.
Saudi Arabia said the attacks were carried out with Iranian weapons, adding that it was capable of responding forcefully and urging UN experts to help investigate the raid.
Crown Prince Mohammed bin Salman said Iranian threats were not only directed against the kingdom but against the Middle East and the world.
While the prince did not directly accuse Tehran, a Foreign Ministry statement reported him as calling on the international community to condemn whoever was behind the strike.
"The kingdom is capable of defending its land and people and responding forcefully to those attacks," the statement added.
Saudi Arabia and Iran have been enemies for decades and are fighting a number of proxy wars.
Trump said he had not made commitments to protect the Saudis.
"No, I haven't promised Saudis that. We have to sit down with the Saudis and work something out," he said. "That was an attack on Saudi Arabia, and that wasn't an attack on us. But we would certainly help them."
Two sources briefed on state oil company Saudi Aramco's operations told Reuters it might take months for Saudi oil production to return to normal. Earlier estimates had suggested it could take weeks.
Saudi Arabia said it would be able to meet oil customers' demand from its ample storage, although some deliveries had been disrupted. At least 11 supertankers were waiting to load oil cargoes from Saudi ports, ship tracking data showed on Monday.
Tension in the oil-producing Gulf region has dramatically escalated this year after Trump imposed severe US sanctions on Iran aimed at halting its oil exports altogether.
For months, Iranian officials have issued veiled threats, saying that if Tehran is blocked from exporting oil, other countries will not be able to do so either. But Iran has denied a role in specific attacks, including bombings of tankers in the Gulf and previous strikes claimed by the Houthis.
US allies in Europe oppose Trump's "maximum pressure" strategy, arguing that it provides no clear mechanism to resolve issues, creating a risk the enemies could stumble into war.
Trump has said his goal is to force Iran to negotiate a tougher agreement and has left open the possibility of talks with Rouhani at an upcoming UN meeting. Iran says there can be no talks until Washington lifts sanctions.
UN Yemen envoy Martin Griffiths told the U.N. Security Council on Monday it was "not entirely clear" who was behind the strike but he said it had increased the chances of a regional conflict.
But the US ambassador to the world body, Kelly Craft, said emerging information on the attacks "indicates that responsibility lies with Iran" and that there is no evidence the attack came from Yemen.
Iran's Yemeni allies have promised more strikes to come. Houthi military spokesman Yahya Sarea said the group carried out Saturday's predawn attack with drones, including some powered by jet engines.
"We assure the Saudi regime that our long arm can reach any place we choose and at the time of our choosing," Sarea tweeted. "We warn companies and foreigners against being near the plants that we struck because they are still in our sights."
US officials say they believe that the attacks came from the opposite direction, possibly from Iran itself rather than Yemen, and may have involved cruise missiles. Wherever the attacks were launched, however, they believe Iran is to blame.
The attacks have raised questions about how the kingdom, one of the world's top spenders on weaponry, much of it supplied by US companies, was unable to protect oil plants from attack.
Sensing a commercial opening, President Vladimir Putin said Russia was ready to help Saudi Arabia by providing Russian-made air defence systems to protect Saudi infrastructure.
Russia and China said it was wrong to jump to conclusions about who was to blame for the attack on Saudi Arabia.