Shipping data showed that the Liberian-flagged merchant vessel Alnic MC was also on its way to Singapore when the vessels collided east of the Strait of Malacca at 5:24 am local time, while it was still dark.
The 885km-wide strait runs between the Malaysian Peninsula and the Indonesian island of Sumatra, connecting the Pacific and Indian oceans. It is one of the world's busiest shipping lanes, but is also well traversed and well governed, analysts say.
The Alnic is more than three times the size of the McCain, with a gross tonnage of 30,000 tons.
Initial reports indicated that the destroyer sustained damage to its port side at the rear, but is currently sailing under its own power and heading to port in Singapore.
The Navy's 7th Fleet said that Navy Seahawk helicopters and Ospreys had been mobilised for the search-and-rescue effort, joining tugboats from Singapore, a Singapore navy ship and helicopters, and a Singapore police coast guard vessel.
"Our first priority is determining the safety of the ship and crew," Admiral John Richardson, chief of naval operations, wrote via Twitter. "As more information is learned, we will share it."
The 7th Fleet set up an emergency assistance center in Yokosuka for family members of the McCain crew.
President Donald Trump, returning to the White House, responded to reporters' questions about the collision by saying: "That's too bad".
This collision comes just days after the Navy issued a damning report listing errors that led to a collision between the USS Fitzgerald - also a Yokosuka-based Arleigh Burke-class destroyer - and a much larger container ship just south of Japan in June.
The collision killed seven sailors, all of whom drowned in their berth compartments when the container ship struck the destroyer's side.
The Navy said last week that it would discipline a dozen sailors who were aboard the Fitzgerald, including the top two officers and the top enlisted sailor, whose careers are almost certainly over.
Admiral William Moran, the vice-chief of naval operations, said that the sailors who were on watch in the ship's bridge "lost situational awareness," contributing to the collision.
Analysts were incredulous that a second destroyer based at Yokosuka could be involved in another collision so soon after the Fitzgerald incident.
Euan Graham, director of the international security programme at the Lowy Institute in Sydney, called it "extraordinary" and said it would certainly lead to pointed questions being asked within the Navy.
But the incident also could affect the 7th Fleet and its readiness, Graham said.
"They were already stretched after the Fitzgerald collision and now they've lost a second frontline destroyer at an acute time in the region, with the tensions around North Korea and in the South China Sea," he said.
The Fitzgerald is still being repaired but is expected to return to service.
The USS McCain is named after the father and grandfather of Senator John McCain. John S. McCain snr and John S. McCain jnr served as admirals in the Navy during World War II. The destroyer's motto is "Fortune favours the brave".
Photos posted on the USS McCain's Facebook page showed the crew enjoying fishing and playing cards on the deck at sunset at the weekend, during a "patrol in the South China Sea in support of security and stability in the Indo-Asia-Pacific region".
- additional reporting AAP
