"The extent of damage and personnel injuries is being determined," it said, adding that the incident was being investigated.
The destroyer was currently sailing under its own power and heading to port.
Senator John McCain tweeted: "Cindy & I are keeping America's sailors aboard the USS John S McCain in our prayers tonight - appreciate the work of search & rescue crews."
The Alnic MC is a 183 metre oil tanker with a deadweight of more than 50,000 tonnes, according to the Marine Traffic website.
This collision comes just days after the Navy issued a report listing a string of 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 berths when the container ship struck the destroyer's side.
The Navy last week said it would discipline a dozen sailors who were aboard the Fitzgerald at the time, including the top two officers and the top enlisted sailor, whose careers are almost certainly over.
Adminstrator William F Moran, the vice chief of naval operations, said the sailors who were on watch in the ship's bridge "lost situational awareness", contributing to the collision.
- Daily Telegraph UK, Washington Post