The United States has described the stopovers as routine and no reason for China to take "provocative" action.
Neither Taiwan nor the United States have given details of Lai's schedule on his stopovers.
That includes the prospect of a visit to the United States by Chinese Foreign Minister Wang Yi, which could pave the way for a meeting between U.S. President Joe Biden and Chinese President Xi Jinping.
Taiwan believes the scale of the exercises could be smaller than those in April, the Taiwanese official said.
Chinese state television this month ran an eight-part series on the People's Liberation Army, some of which focused on Taiwan.
Persons:
William Lai's stopovers, Lai, Tsai Ing, Kevin McCarthy, Biden, Wang Yi, Joe Biden, Xi Jinping, China, Jeff Liu, Yimou Lee, Ben Blanchard, Liz Lee, Michael Martina, Robert Birsel
Organizations:
United, House, Taiwan Affairs Office, People's Liberation Army, China's Eastern Theatre Command, Reuters, Chinese Foreign, American Institute, The U.S . State Department, Thomson
Locations:
TAIPEI, China, Taiwan, United States, U.S, Beijing, Los Angeles, Taiwan Strait, New York, Paraguay, San Francisco, stoke, Virginia, The, Taipei, Washington