Former South Korean President Park Geun-hye arriving in court in Seoul last month. She was impeached on corruption charges in March. Credit Kim Hong-Ji/Reuters

SEOUL, South Korea — North Korea issued standing orders on Wednesday for the “miserable dog’s death” execution of South Korea’s imprisoned former president and her spy chief, and improbably demanded that its southern adversary extradite them.

The execution orders, which the North said could be carried out anytime, anywhere and by any means, amounted to an assassination decree against the imprisoned former president, Park Geun-hye, and Lee Byung-ho, who was director of the National Intelligence Service under Ms. Park.

Conveyed in a statement issued via North Korea’s official news agency, the execution orders came nearly two months after the isolated, nuclear-armed country accused the South Korean intelligence service of conspiring with the C.I.A. to assassinate the North Korean leader, Kim Jong-un, using biochemical poisons.

“We declare at home and abroad that we will impose death penalty on traitor Park Geun-hye and Mr. Lee,” said the statement from the North’s Ministry of State Security, Ministry of People’s Security and Central Public Prosecutors Office. The statement said the pair’s crime was “hideous state-sponsored terrorism.”

Ms. Park and Mr. Lee “can never make any appeal even though they meet a miserable dog’s death anytime, at any place and by whatever methods from this moment,” the statement said.

There was no immediate response from Ms. Park, who has been incarcerated since she was forced out on corruption charges in March, or from the National Intelligence Service. But the South Korea spy agency has denied trying to assassinate Mr. Kim.

North Korea said Ms. Park had endorsed an “operation to replace the supreme leadership of the North” since late 2015. It said the plot involved Mr. Kim’s assassination, which it said South Korean agents had planned to disguise as a “car or train accident.” But the plot was eventually abandoned after Ms. Park was impeached in a corruption scandal, North Korea said.

The statement provided no further details of the supposed plot. But on Monday, the Japanese newspaper Asahi, quoting anonymous sources, reported that Ms. Park had approved a plan for the National Intelligence Service to overturn Mr. Kim’s government in 2015.

The spy agency has called the report groundless.

Ms. Park’s truncated term in office was punctuated by a spike in tension with North Korea, which had once tried to assassinate her father when he was the dictator of South Korea nearly a half-century ago.

Mr. Kim has accelerated the North’s nuclear weapons program with a torrent of bomb and ballistic missile tests, threatening South Korea and the United States with a “nuclear sword of justice.”

Ms. Park was an advocate of tough sanctions against the North while it fulminated, calling her a “snake” and a “prostitute.”

Mr. Kim, grandson of North Korea’s founder, Kim Il-sung, runs an autocracy especially sensitive to any hint of threat to its leader. North Korea is widely believed to have been responsible for the brazen hacking of Sony Pictures in 2014 as retaliation for “The Interview,” a fictional American film centered around a plot to assassinate Mr. Kim. Although North Korea is known to have tried to assassinate South Korean leaders at least twice, it is highly unusual for the North to claim to be a victim of a South Korean plot.

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