US military commanders have warned the Russian Navy and Russian Air Force to act in a professional and safe fashion when the fleet of 32 ships deploys into the region.
There are now concerns that the Black Sea could become a flashpoint with Moscow.
Russia's Defence Ministry claimed one of its vessels had fired warning shots and a warplane had dropped bombs in the path of a Royal Navy destroyer last week.
President Vladimir Putin suggested on Wednesday that if his military had sunk HMS Defender it would have been “unlikely” to start the Third World War.
There will be increased international focus on Exercise Sea Breeze 21 when its warships, led by the advanced missile-destroyer USS Ross, leaves the Ukrainian port of Odessa and steam eastward into the Black Sea.
The fleet is expected to pass through the waters off Crimea that Russia claims to be its own after it seized the peninsula in 2014.
When HMS Defender sailed within 22 kilometres of Crimea last Thursday, the Russians claimed to have fired warning shots. The British Ministry of Defence rejected the statement, adding its ship was sailing through international waters.
Mr Putin did nothing to defuse tensions on Wednesday when he made a quip about the incident during his annual live broadcast interview with the Russian people.
“Even if we had sunk the British destroyer near Crimea it is unlikely that the world would have been on the verge of World War Three,” he said.
He stuck to the Russian position that the advanced Type-45 destroyer had illegally entered territorial waters near Crimea to observe in detail how Russian forces would react
“This was a provocation, of course,” Mr Putin said. “It was obvious that the destroyer entered pursuing, first of all, military goals, trying … to see what happens on our side, how things work and where everything is located.”
Whether the Russian Black Sea fleet, based in Sevastopol, Crimea, decides to take the same belligerent stance against the American and other Nato and foreign warships will be a key moment.
An aggressive move would undermine the efforts made by President Joe Biden to find a working relationship with the Kremlin following his Geneva meeting with Mr Putin earlier this month.

The American fleet commanders appeared in no mood to compromise on Crimea territorial waters during a remote media briefing with the media yesterday.
Asked if the fleet would avoid international waters not recognised by the Russians, Captain Kyle Gantt responded: “An important part of what we’re doing here with Sea Breeze is we are demonstrating to the world that the Black Sea is an international sea; it is open and available for the free transport of commerce, of shipping, for all nations, and it is not owned by any one nation.”
He added that there was a longstanding process in place for communications with Russia, called the Incidents at Sea Agreement, that “removes risk”.
Pressed on the potential Russian response, given its hostility to the Royal Navy, Captain Gantt, deputy commander of Task Force 65, said he expected all nations to “operate professionally” at sea.
“We expect that all nations will respect the international laws and norms that govern safe and professional operations at sea. I expect that from Russia.”
The exercise has the largest number of participating nations in its history with 32 countries providing 5,000 troops, 32 ships and 40 aircraft.
The plan will also include a joint US Navy and Ukraine operation to remove the wreck of a Soviet-era Ministry of Interior yacht that sank off Odessa’s naval pier in 2016.
Its removal will allow Ukraine’s navy to operate with greater freedom in the Black Sea.