As North Korea races to build a weapon that for the first time could threaten American cities, its neighbors are debating whether they need their own nuclear arsenals.

The North’s rapidly advancing capabilities have scrambled military calculations across the region, and doubts are growing the United States will be able to keep the atomic genie in the bottle.

For the first time in recent memory, there is a daily argument raging in both South Korea and Japan — sometimes in public, more often in private — about the nuclear option, driven by worry that the United States might hesitate to defend the countries if doing so might provoke a missile launched from the North at Los Angeles or Washington.

In South Korea, polls show 60 percent of the population favors building nuclear weapons. And nearly 70 percent want the United States to reintroduce tactical nuclear weapons for battlefield use, which were withdrawn a quarter-century ago.