ISTANBUL—A U.S. congressional vote on how to define the 1915 slaughter of Armenians in the Ottoman Empire, expected Thursday, is turning into a game of brinksmanship between the White House and Ankara.

In previous years, Congress has attempted to pass a resolution to recognize the Armenian events as genocide. Such a resolution would inflame Turkey and has brought vows from past U.S. administrations that they would block the bill, a nod to Turkey's role as a key ally of Washington in the Middle East.

This...