The USS John S. McCain is still shadowing the Kang Nam -- the North Korean vessel suspected of carrying weapons or nuclear material -- and it is currently 200 miles south of Shanghai, according to a senior U.S. defense official.
The ship is being watched very closely by the U.S. military for indications as to where it will reach port.
The working hypothesis in the military intelligence community is that it is heading to Burma, the military dictatorship also known as Myanmar, and could make its way through the Strait of Malacca and possibly have to refuel in Singapore.
But that is not a certainty, according to a U.S. military official, just a working assumption. It is moving very slowly -- about 10 knots.
U.S. military does not know when it will run out of fuel, and according to one official, it is "not a very fuel efficient vessel" so it is difficult to extrapolate when it will need to refuel.
FOX News' Jennifer Griffin and Justin Fishel contributed to this report.