Bernhard Schulte Ship Management, which operates Kokuka Courageous, said it launched an emergency response following a security incident and added that the vessel was not at risk of sinking and its cargo of methanol was intact.
The vessel was travelling from Al Jubail in Saudi Arabia to Singapore, according to ship-tracking website MarineTraffic.
The US Fifth Fleet said it was assisting the stricken vessels in the Gulf of Oman, a key waterway used to transport energy supplies from the Middle East.
A second tanker, Norwegian-owned Front Altair, had set sail from the Emirati port of Ruwais late on Tuesday and was set to arrive at the Taiwanese port of Kaohsiung on June 30. The tanker suffered a fire following an explosion while sailing in the Gulf of Oman, vessel manager Frontline said.
"We are aware of the reported attack on shipping vessels in the Gulf of Oman," the Bahrain-based US 5th Fleet said in a statement.
"US Naval Forces in the region received two separate distress calls at 6.12 a.m. (Bahrain) time and a second one at 7 am," it said, adding that the US Navy ships were in the area and rendering assistance.
The cause of the blasts in one of the world's busiest oil routes was unclear and both vessels were still afloat.
The Iranian rescue ship Naji 10 picked up 44 crew members from the tankers, 23 from one freighter and 21 from the other, and took them to Bandar-e-Jask, in the southern Iranian province of Hormozgan, the official IRNA news agency reported, later adding that other ships passing nearby had also taken part in the rescue.
"Reported attacks on Japan-related tankers occurred while Prime Minister Shinzo Abe was meeting with (Iranian Supreme Leader) Ayatollah Khamenei for extensive and friendly talks," Iran's Foreign Minister Javad Zarif said.
"Suspicious doesn't begin to describe what likely transpired this morning," he added.
The UK's Maritime Trade Operations said it was investigating the alleged attack on the two tankers. Japanese Minister of Economy, Trade and Industry Hiroshige Seko also made a similar statement.
The incident came as Japanese Prime Minister Shinzo Abe met senior Iranian officials in Tehran on Wednesday during a visit that aimed to de-escalate soaring tensions between the US and Iran.
The alleged attack on Thursday came almost exactly one month after four commercial ships were purportedly sabotaged by mines off the port of Fujairah. The US laid the blame for the incident firmly on Iran or Iranian proxies.
The Gulf of Oman is connected to the Persian Gulf by the Strait of Hormuz, which has become the geopolitical focal point in rising tensions between the US, its Arab allies and Iran since Donald Trump's administration withdrew from the 2015 nuclear deal with Tehran and re-applied economic sanctions.