A wide-ranging search and rescue operation is under way. It is the second collision involving a US Navy ship in recent months.
What do we know about the collision?
It was reported before dawn at 05:24 local time on Monday (21:24 GMT on Sunday) and took place east of the Strait of Singapore, as the USS John S McCain prepared to perform a routine port stop in Singapore.
Singapore and US authorities said the destroyer sustained damage to her port side, which is the left-hand side of the vessel facing forward.
The tanker it collided with, Alnic MC, sustained damage to a tank near the front of the ship 7m (23ft) above the waterline, but none of its crew were injured and there were no oil spills.
The tanker has a displacement of 50,700 tons, making it almost six times heavier than the USS John S McCain, and at 182m (600ft) it is slightly longer than the 154m long US destroyer.
What is happening now?
US military helicopters as well as the Singaporean and Malaysian navies and coast guards are currently conducting search and rescue operations.
Malaysian Navy chief Admiral Kamarulzaman said a message had been sent to fishing vessels along the Johor and Pahang coasts to keep a look out for missing sailors.
A Malaysian Navy spokesman later told a news conference: "The waves are between 0.5m and 1.5m, so the sea is quite rough and can be very challenging if you don't have a life jacket."
The US Navy said four of the injured sailors had non-life threatening injuries and were medically evacuated to a Singapore hospital. The fifth person did not require further medical attention.
The US Navy 7th Fleet later issued a statement saying the USS John S McCain had now arrived at Singapore's Changi naval base.
It said: "Significant damage to the hull resulted in flooding to nearby compartments, including crew berthing, machinery, and communications rooms. Damage control efforts by the crew halted further flooding."
The Alnic MC is making its way to the Raffles Reserved Anchorage in Singapore.
One crew member of the oil tanker told Reuters over the phone that the vessel sustained some damage to a valve. The ship was carrying nearly 12,000 tonnes of oil from Taiwan to Singapore, the news agency reported.
The Maritime and Port Authority of Singapore said: "There is no report of oil pollution and traffic in the Singapore Strait is unaffected."