The second ship, a Japanese-owned tanker, was hit by a suspected torpedo, the firm that chartered the ship said. Its crew were also picked up safely. However, a person with knowledge of the matter said the attacks did not use torpedoes.
The Bahrain-based U.S. Navy Fifth Fleet said it had assisted the two tankers after receiving distress calls.
Iran has not openly acted on its threat to close the Strait of Hormuz even though U.S. sanctions have seen its oil exports drop from 2.5 million barrels per day in April last year to around 400,000 bpd in May.
Both sides have said they want to avoid war.
Bob McNally, president of the U.S. consultancy Rapid -- an Energy Group, said "we see this as Iran trying to get negotiating leverage it doesn't have," and described the attacks as "upping the ante but not going all in."
"I don't think it tips us over into direct military confrontation. It is still deniable and denied. This is still going to be like the attack last month everyone is denying it. It's a blunt message."
Japanese Prime Minister Shinzo Abe was visiting Tehran when Thursday's attacks occurred, carrying a message for Iran from Trump, who has demanded that the Islamic Republic curb its military programs and its influence in the Middle East.
Abe, whose country was a big importer of Iranian oil until Washington ratcheted up sanctions, urged all sides not to let tensions in the area escalate.
Iran said it would not respond to Trump's overture, the substance of which was not made public.
Britain said it was "deeply concerned" about the attacks. Germany, which like Britain remains a signatory to the nuclear pact with Iran, said the "situation is dangerous" and all sides needed to avoid an escalation.
The Arab League said some parties were "trying to instigate fires in the region," without naming a particular party.
Oman and the United Arab Emirates, which have coastlines on the Gulf of Oman, did not immediately issue any public comment.
Saudi Arabia and the UAE, both majority Sunni Muslim nations that have a long-running rivalry with predominantly Shi'ite Iran, have previously said attacks on oil assets in the Gulf pose a risk to global oil supplies and regional security.
Bernhard Schulte Shipmanagement said the Japanese tanker Kokuka Courageous was damaged in a "suspected attack" that breached the hull above the water line while transporting methanol from Saudi Arabia to Singapore.