The crew aboard the Donald Cook made several attempts to radio the Russian warplane asking the pilot what were his intentions and sending warnings to remain at a safe distance, but the Russian pilot never responded.
The incident is reported to have occurred yesterday and the US is expected to lodge a protest with Russia against what the Pentagon alleged as the provocative and unprofessional behaviour of Russian fighter jets.
"It is provocative and unprofessional. Russian action is inconsistent with the international protocol, and various agreements on professional interactions between our militaries," the Pentagon spokesperson said, adding that at no point of time the American naval ship was in danger. The Russian ships were unarmed, he noted.
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"The Russian plane made a total of 12 passes. It did not overfly the deck of the Donald Cook," he said, adding that the Russian fighter jet was not exactly circling over the US ship.
There was another Russian fighter jet (S-24) which was at a very high altitude.
"There were a total of two aircraft. Only one (of them) made these lower-level passes. The second one remained at much higher level passes.
Responding to questions, the Pentagon spokesperson said USS Donald Cook remains prepared to defend itself and made several attempts to contact this Russian plane. So far no protest has been lodged so far, but the issue would be taken up by the State Department, he said.
Secretary of State John Kerry, is scheduled to meet his Russian counterpart Sergei Lavrov in Europe later this week.
There has been no communication on this issue between the militaries of the two countries, he added.