Details of the new deployments were included in Greenert's "Position Report" issued on Tuesday.
Along with those new ships, Greenert is permanently stationing four more Navy minesweeping crews in Bahrain, "complemented by new minesweeping systems that expand their capability," according to the Navy chief.
In the Pacific, the Navy is moving its newest warship to Singapore as part of the Pentagon's strategic shift to the Pacific.
The USS Freedom, the first Littoral Combat Ship in the Navy fleet, will be sent to the Asian nation to help patrol the Strait of Malacca.
The waterway is the main thoroughfare for military and commercial vessels moving between the Pacific and Indian oceans, with nearly half of the world's global trade traversing the straits annually.
The increased minesweeping capability heading to Fifth Fleet comes less than a month after a massive, Navy-led multinational exercise off the coast of Iran designed to hone the sea service's mine-hunting skills.
Navy carrier strike groups USS Stennis and USS Eisenhower led the main American naval force during the exercise, which also including NATO, the United Kingdom, France, Saudi Arabia and United Arab Emirates.
The goal, according to Navy leaders, was to sharpen current tactics and develop new ones to counter sea mines, a favorite weapon of the Iranian navy.
Tehran has littered the coastal waters along the Straits of Hormuz and elsewhere in the Persian Gulf with mines, as part of the game of military one-upmanship being played between Washington and Iran over the critical waterways in the Mideast.
But in spite of that large U.S. and international buy-in, the exercise was less than successful with the U.S. and allies reportedly only being able to detect half of the decoy mines used during the drill.
That said, the Navy continues to pursue increased military partnerships with various nations in the Mideast.
Talks are under way between senior Navy officials and their counterparts in Cairo to begin conducting joint war games for the first time since Egyptian president Mohamed Morsi took power earlier this year.
The tentative exercises would focus on improving Egypt's ability to deal with small boat attacks and general patrol operations of its coastal waters, Greenert told reporters in September.
In the near term, the Pentagon is laying out the conditions with the country's new military leaders to allow American warships to begin docking at Egyptian ports again, Greenert said during a speech in Washington.