He might be South Korea’s next president. All he has to do is impeach the current one.

Now, a short-lived attempt by South Korean President Yoon Suk Yeol last week to impose martial law has thrust Lee to within striking distance of his nation’s top job, as Yoon faces impeachment efforts and members of his cabinet are under criminal investigation.

Lee, who narrowly lost to Yoon in presidential elections in 2022, declined to say whether he would seek the office again. But in an interview, the 59-year-old former labor-rights lawyer said he was determined to oust Yoon, whom he described as a threat to Korea’s constitutional order.

“We have to impeach him and restore normal democracy," Lee told The Wall Street Journal, warning that as long as Yoon and his backers stay in power there is a risk they will try to delcare martial law again.

Lawmakers last week failed to reach a voting quorum to impeach Yoon for his martial-law decree. An impeachment motion filed by Lee’s left-leaning Democratic Party, which controls a majority of seats in the National Assembly, fell through on Saturday, when representatives of Yoon’s People Power Party abstained.

Opinion polls indicate that Lee would come out on top in a snap election, which would be called if Yoon is removed or resigns. One survey released Monday found that 52% support Lee. About 10% said they would vote for Han Dong-hoon, the leader of the People Power Party.

The ruling party, whose own popularity is sinking, worries it would struggle to fend off Lee in the event of a Yoon departure, so it is trying to stall. On Sunday, Han said Yoon would be politically sidelined and uninvolved in running the country or diplomacy until an unspecified “orderly early retreat."

Instead, Han said, South Korea’s “livelihood issues and state affairs" would be jointly handled by the ruling party and the country’s prime minister.

Lee said that this arrangement amounts to a “second act of insurrection" by Han and the ruling party. He condemned it as unconstitutional and likely worthy of criminal punishment. “The president is elected by the people—not the People Power Party," Lee said.

On Monday, Yoon was placed under a travel ban while he is investigated for charges that include insurrection—one of the few offenses that isn’t protected by presidential immunity. South Korea’s military also clarified the same day that Yoon, despite being sidelined, would legally remain the country’s commander in chief.

Yoon, about halfway through a five-year term set to end in 2027, has admitted he called for martial law last week out of desperation, citing fears his political enemies had made the country vulnerable to North Korean “communist forces."

‘Warlike politics’

South Korean politics is highly polarized. In January, Lee was stabbed in the neck by a politically motivated assailant who wanted to stop him from one day becoming president. Emergency surgery saved his life. Afterward, he called for an end to the country’s “warlike politics." But tensions between the political right and left remain high.

Lee has five ongoing trials for alleged offenses including perjury, breach of duty and involvement in illicit fund transfers made to North Korea. Last month, a court found him guilty of election-law violations, a verdict he has appealed. Lee has denied any wrongdoing. If he is ultimately convicted in the election-law violation case, he would lose his parliamentary seat and be unable to run for office for 10 years after the conviction.

For Lee, South Korea’s brief return to martial law evoked a period in May 1980, when he was a 15-year-old factory worker in a Seoul suburb who had quit school to help his family make ends meet. He recalled thinking then that the state—and the military dictator that led it—were justified in taking on protesters, who were portrayed as having stolen weapons and shot at soldiers.

Two years later, after passing a special exam to enter college, Lee learned from fellow students and others the truth about Gwangju: The citizens who rose up against military rule weren’t rebels, Lee said, but victims shot by the military first. He said the realization spurred him to eventually seek public office.

“I decided to dedicate myself to creating a world where such things couldn’t happen," Lee said.

That is part of the reason why Lee sprung into action Tuesday night. Not long after Yoon finished his speech declaring martial law, Lee sent out a Telegram message at 10:39 p.m. to the Democratic Party leadership to gather quickly.

As his wife drove him to the National Assembly from their home in the Seoul suburbs, Lee livestreamed his entrance to his more than 1 million YouTube subscribers, urging citizens to flood the legislative compound in protest.

‘I am a realist’

In the past, Lee’s critics called him “South Korea’s Bernie Sanders," owing to offering a universal basic income to young people in the province he governed, among other progressive policies. He didn’t shy away from the comparison either. More recently, due to his legal problems, passionate base and social-media presence, a different comparison has emerged. “Some have even said I’m like the ‘Trump of Korea,’ " he said.

Lee said he doesn’t see himself as hyperpartisan, calling himself a pragmatist. “I am a realist," he said.

He would represent a break from the current conservative Yoon administration, which has become more confrontational with North Korea, has strengthened ties with Japan and has fortified relations with the U.S.—including providing indirect aid to Ukraine by selling shells to America.

Lee said Yoon “keeps wanting to get dragged" into the Ukraine war, a reference to Seoul leaving open the possibility of sending further aid to the country since North Korea sent soldiers to fight alongside the Russians. While Yoon had vowed South Korea won’t “sit by idly," Lee has called aid to any foreign country a national-security risk.

Lee “highly values" Trump’s stated goal of ending the Ukraine war, as well as Trump’s apparent interest in engaging again with Kim Jong Un.

“We cannot help but be very grateful," Lee said. “President-elect Trump has attempted things that others find difficult."

Lee said Yoon has unnecessarily heightened tensions with China.

To get the chance to lead South Korea, Lee will need Yoon to either resign or sway more of Yoon’s ruling-party lawmakers to break ranks. The opposition controls 192 seats in South Korea’s unicameral 300-seat National Assembly. They need a two-thirds majority to pass impeachment.

All the opposition needs, Lee said, is for eight members to cross the aisle.

“When the water exceeds the limit, it will overflow quickly. And you will choose to live together rather than die," Lee said.

Soobin Kim contributed to this article.

Write to Timothy W. Martin at Timothy.Martin@wsj.com, Gordon Fairclough at Gordon.Fairclough@wsj.com and Jiyoung Sohn at jiyoung.sohn@wsj.com