Oil prices have climbed 3.4% since Thursday's attacks. Ship insurers said insurance costs for ships sailing through the Middle East have jumped by at least 10%, Reuters reported.
Saudi Arabia's energy minister said in Japan at a meeting of energy ministers from the G20 group of nations that the kingdom is committed to ensuring stability of global oil markets.
Japanese Industry Minister Hiroshige Seko said ministers agreed on the need to "work together to deal with the recent incidents from (an) energy security point of view."
Trump, who pulled the United States out of the nuclear deal under which world powers agreed to ease international sanctions on Iran in return for curbs on Tehran's nuclear work, said any move to close the Strait of Hormuz would not last long.
He also said he was open to holding talks with Iran, although Tehran said it had no plans to negotiate with the United States unless it reversed a decision on the nuclear deal.
Tehran and Washington have both said they have no interest in a war. But this has done little to assuage concerns that the arch foes could stumble into conflict.
A US official told Reuters a surface-to-air missile was fired from Iranian territory on Thursday morning at a US drone that was near Front Altair following the attack on the tanker. The missile did not hit the drone, the official said.
Acting US Defence Secretary Patrick Shanahan said the United States was "planning various contingencies" when asked if more military forces would be sent to the area, but added that the focus was on building an international consensus.
"We also need to broaden our support for this international situation," he told reporters on Friday.