WASHINGTON — The last time a former American president traveled to North Korea on a rescue mission — Bill Clinton, a year ago — he was feted by its leader, Kim Jong Il, who seized on the visit to reach out to the Obama administration. This week, Kim chose to go to Chinaduring a visit by former President Jimmy Carter to free another jailed American.

Whatever the motivation for Kim’s snub, analysts said it underscored the deep freeze between North Korea and the United States. The State Department greeted the news Friday that Carter had secured the release of Aijalon Mahli Gomesby warning other Americans not to go to North Korea, saying they risked "heavy fines and long prison sentences with hard labor."

Even as it keeps up its tough tone, however, the United States has begun weighing a fresh effort at engagement with Kim’s government, officials and analysts briefed on the deliberations say.

Such an overture would come "several moves down the chessboard," a senior official said, and would be preceded by additional pressure tactics. But it suggests that the administration has concluded that pressure alone will not be enough to move North Korea’s ailing, reclusive dictator.

At a high-level meeting last week on North Korea, Secretary of State Hillary Rodham Clinton solicited ideas from outside experts and former officials about the next steps in policy toward the North. The consensus, even among the hawks, was that the United States needed to resume some form of contact with Kim, according to several people who took part.

Clinton, these people said, expressed impatience with the current policy, which is based on ever more stringent economic sanctions and joint American-South Korean naval exercises — both in response to the sinking in March of a South Korean warship, for which South Korea blamed the North.

Among those advocating a fresh overture is Stephen W. Bosworth, the special envoy for North Korea. He visited Pyongyang, the North’s capital, in December to explore the prospect of talks, but the administration could not decide whether to schedule a follow-up meeting, and then the warship was torpedoed.

"The question is, what are we going to do now?" said Joel S. Wit, a former State Department negotiator with North Korea who founded a website, 38 North, which follows North Korean politics. "The answer is re-engagement. There aren’t any other tools in the toolbox."

Far from abandoning pressure tactics, officials said, the United States is likely to increase them. In July, it announced new measures aimed at choking off sources of hard currency for the government and its allies. Clinton sent a senior adviser, Robert J. Einhorn, to Asia to drum up support for the sanctions. The military, defying threats from North Korea and anger from China, has held several days of joint drills with South Korea in the Yellow Sea.

But concern is growing, even among hawkish analysts, that pressure, without any dialogue, raises the risk of war. Some critics also contend that there is little evidence the sanctions have forced the North to retreat from its nuclear program or its belligerence toward South Korea.

Kim’s deteriorating health, and the succession struggle it has set off, have increased the pressure on the administration to reach out, in the view of some analysts. While some officials argue that the United States can wait out the political transition, others fear that heightening the confrontation with North Korea could foreclose future opportunities for contact.

As Victor Cha, a former Bush administration official who was responsible for North Korea, put it, "If they look like they’re preparing for war, there’s no opportunity to talk to the new leadership."

The administration, analysts said, is also losing confidence in China’s willingness to press the North. During a visit to Beijing in May, Clinton invested a lot of energy in trying to persuade Chinese officials to accept the South Korean government’s finding that the North had sunk its ship. Her efforts were futile: Beijing never accepted the North’s culpability and it blunted Seoul’s drive for a U.N. statement condemning the attack.

Symbolically, analysts said, Kim’s choice of a trip to China over a meeting with Carter highlighted North Korea’s economic and political dependence on Beijing. China has long pushed for the United States to talk to the North, and reopening a dialogue could help ease the tension between Beijing and Washington over issues like the military exercises.

One problem for the administration is the form and content of talks. Few analysts have much enthusiasm for the six-party format, under which North Korea has negotiated over its nuclear program with the United States, South Korea, China, Japan and Russia. But the talks are probably necessary to retain support of allies like South Korea and Japan.

Another problem is that the administration has been uncompromising in its demands. Officials have repeatedly said that the United States will not negotiate until North Korea agrees to dismantle its nuclear weapons. Their fear is that the North will extract concessions, as it did during the Bush and Clinton administrations, only to test another nuclear bomb.

"We don’t want to go down the old road and repeat the experiences of the past," said a senior official, speaking on the condition of anonymity because he was not authorized to speak publicly. An option, experts said, would be to engage North Korea on issues other than the nuclear program. But others said the nuclear issue was unavoidable.

For now, the administration offers a more pragmatic strategy. "Americans should heed our travel warning and avoid North Korea," said the State Department’s spokesman, Philip J. Crowley. "We only have a handful of former presidents."