SEOUL—North Korea’s nuclear push is triggering a military buildup here and adding fuel to a hot debate over South Korea’s defense strategy—including whether the country should have its own nuclear option.

A few conservative politicians and a small majority in opinion polls have for years supported South Korea getting access to nuclear weapons. Lately, some prominent new voices have joined them, including Kim Jin-pyo, a four-term lawmaker from the main, left-of-center opposition party, who said Seoul needed a “balance of terror” to match North Korea’s threat.

Mr. Kim said nuclear weapons in South Korea would also pressure China and Russia to deal with North Korea more seriously.

Two of South Korea’s major newspapers have recently shifted from caution to strongly supporting the idea of basing U.S. nuclear weapons in South Korea again, 25 years after they were removed in favor of a strategy of extended deterrence.

In a September report, a panel of experts that advises South Korean President Park Geun-hye on policies to reunify the Korean Peninsula also suggested studying the reintroduction of U.S. nuclear weapons.