Jenkins, accused of crossing the Demilitarized Zone and defecting to North Korea in 1965, will be hospitalized in Tokyo on Sunday after he arrives with his Japanese wife and their two daughters, Chief Cabinet Secretary Hiroyuki Hosoda said at a news conference.
Bringing Jenkins to Japan is a top political priority for Tokyo. He has been in Indonesia for the last few days meeting with his family.
A North Korean official who accompanied Jenkins to Indonesia told reporters he wouldn't stand in the way of Jenkins traveling to Japan.
"Wherever the family decides to live, we will respect their wishes," Japanese public broadcaster NHK showed one of the officials telling reporters in Jakarta. "We hope Jenkins recovers."
The official spoke after meeting with Jenkins and his wife at their hotel.
Jenkins' wife, Hitomi Soga, was kidnapped and taken to the North by communist agents in 1978. She wasn't allowed to return to her homeland until 2002, when Pyongyang admitted it had abducted more than a dozen Japanese citizens in the 1970s and 1980s to teach Japanese to its spies.
Since then, Soga has been living separated from her family because Jenkins declined to leave North Korea over fears the United States would demand his extradition to face desertion charges. The United States may not ask for custody while he is hospitalized for medical treatment, however.
Soga says she wants to live in Japan with her family.
"Jenkins must be treated so he can recover soon," Prime Minister Junichiro Koizumi told reporters Friday. "There is no change to our policy of enabling Soga's family to live in Japan together."
Jenkins had earlier refused to go to Japan because of his fears he would be extradited. But Kyodo News, citing unidentified Japanese officials in Jakarta, reported that Jenkins told the Northern officials on Friday that he would go to Japan and gave them the North Korean cash he had with him, saying he wouldn't need it anymore.
Japanese doctors sent by the government to examine Jenkins in Indonesia recommended he be sent to Japan for further care. Hosoda said Jenkins is suffering from problems following abdominal surgery in North Korea.
U.S. Ambassador to Japan Howard Baker said Jenkins is still classified as a deserter, and when the U.S. can gain custody of him, he will be charged, reports CBS News' Jason Testar. He could face life in prison if convicted.
On Thursday, officials in Washington reiterated that the United States would pursue its case against Jenkins.
"Once he is in Japan, he ... falls under the authority of the U.S. military," State Department spokesman Richard Boucher said in Washington. "We intend to request custody when we have the legal opportunity to do so."
Jenkins, 64, has apparently never been processed out of the military and presumably would be subject to U.S. military arrest and court-martial in Japan, where some 50,000 U.S. troops are based under a mutual security pact.
Neither Koizumi nor Hosoda would directly comment on whether Jenkins should turn himself in to U.S. authorities. "This is a matter that must be considered from many angles," Hosoda said.