McCain is now sailing back to port using its own power, according to the Navy.
The US Navy has also sent its V-22 Osprey aircraft and Sikorsky SH-60 Seahawk helicopters from the amphibious assault ship USS America to assist the warship.
The rescue operation involves tug boats, a helicopter and a police coast guard vessel, and U.S. aircraft were also responding, the navy said.
Alnic MC is a 600-foot oil and chemical tanker with a gross tonnage of 30,000.
He followed up with a Tweet: "Thoughts & prayers are w/ our @USNavy sailors aboard the #USSJohnSMcCain where search & rescue efforts are underway".
The US Navy stated that the USS John S. McCain collided with the merchant vessel while the US Navy destroyer was on its way to dock at a harbor in Singapore.
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President Trump's initial response to the accident was "that's too bad".
This is the second incident involving a U.S. Navy vessel in the last two months.
"He has put together a broader inquiry to look into these incidents and to determine any of the causal factors to determine what's going on, both immediate contributors to this, but also any, any related factors", Mattis said.
"There is no report of oil pollution and traffic in the Singapore Strait is unaffected".
"Thoughts & prayers are w/ our @USNavy sailors aboard the #USSJohnSMcCain where search & rescue efforts are underway", Trump said in a post which contained a link to the U.S. military's announcement about the crash.
The US Navy released a statement saying that the USS John McCain, a member of the Navy's 7th Fleet, collided with the Alnic MC oil tanker at 5:24am local time.