The family of Kenneth Bae, the American sentenced to 15 years of hard labor in North Korea, received letters from him in the mail for the first time this past week, telling them that his health is worse and asking them to press the United States government to help secure his release, Mr. Bae’s sister said Friday.

North Korea experts said the message of the handwritten letters — and their method of delivery, which could not have happened without North Korea’s approval — suggested that the authorities there were open to the idea of negotiations on Mr. Bae. That had seemed remote three months ago when he was found guilty of committing “hostile acts” against the government…

The possible opening in Mr. Bae’s case came against a backdrop of other indications that North Korea, despite its harsh public language toward the United States, is pursuing multiple ways of pushing for direct contact after months of threats and new weapons tests. So far, the Obama administration has resisted the overtures.