So far, his efforts to find a new location for the base have not appeased Washington; it initially demanded that Tokyo adhere to the original 2006 deal but has recently signaled greater flexibility. The 2006 deal calls for moving the base from its current location, in the center of the city of Ginowan, to Camp Schwab, an existing Marine base in less-populated northern Okinawa.

Image The Futenma Marine Corps Air Station, in the city of Ginowan, would be moved elsewhere on Okinawa under a 2006 deal. Credit... Toshifumi Kitamura/Agence France-Presse — Getty Images

The perception that Mr. Hatoyama has mishandled the relationship with the United States, Japan’s longtime protector, has contributed to his falling approval ratings, which have dropped below 30 percent. Opposition leaders and media commentators have begun calling on him to resign if he fails to find a compromise by the end of May.

While Mr. Hatoyama has remained tight-lipped about what his plan may look like, officials from his government have made repeated visits to Okinawa to sound out local leaders. Okinawan politicians and the local news media have described the emerging plan as a modified version of the 2006 agreement.

They said the government was considering building a smaller airbase at Camp Schwab than under the 2006 agreement and moving at least part of Futenma’s functions — most likely some of its training operations, and perhaps some of its helicopters — to Tokunoshima, a smaller island about 120 miles north of Okinawa. Japanese news media have interpreted this proposal as a token gesture to appease Okinawans by moving at least some of the Marines off the island.

Okinawan leaders and local media reports have also said the government is considering constructing a new air base on an artificial island to be built off the Okinawan city of Uruma. Japanese media reports have said the island could take decades to build and would serve as a longer-term home for the Marines.