BEIJING/NUSA DUA, Indonesia (Reuters) - China cast doubt on Tuesday on reports that Japanese Prime Minister Shinzo Abe and Chinese President Xi Jinping shook hands at an Asia-Pacific summit, saying the real issue was that Japan had to be sincere and act to end a territorial dispute.

Relations between the world's second- and third-largest economies have been troubled for months because of a sovereignty dispute over tiny islands in the East China Sea known as the Senkaku in Japan and the Diaoyu in China.

Abe and Xi also shook hands on the fringes of the G20 summit in St. Petersburg in September, the first such show of cordiality between the two since they took office.

Tomohiko Taniguchi, a councillor for the cabinet secretariat in Abe's office, said on Monday that the two had shaken hands again at an Asia-Pacific Economic Cooperation (APEC) meeting on the Indonesian island of Bali but Chinese Foreign Ministry spokeswoman Hua Chunying declined to confirm that.

"What I have seen is some Japanese media reporting Xi Jinping and Prime Minister Abe shook hands during an informal leaders meeting at APEC. Don't you think that this kind of report and its hyping up are very boring?" Hua said on Tuesday.

"If you pay attention to the news released by the Chinese delegation you'll discover that there are no reports on this. So what problem does this show?" she told a daily news briefing in Beijing.

Hua repeated that China was willing to talk to Japan, but said Abe was not serious about resolving the dispute over the islands.

"The problem is that Japan has to show sincerity and take real actions to appropriately resolve the relevant issue," she said.

"The Japanese side has not made any real efforts, but has craved hyping up reports of this kind. Do you think this is really interesting? As a responsible leader, he should understand what he needs to do to really improve bilateral relations, not attempt the impossible and put the cart before the horse."

Taniguchi said Abe was open to talks with China that would sidestep the island dispute, which has been exacerbated by the purchase of several of the islands late last year by the Japanese government from a private owner.

"There is just one area of issue that it is pretty hard for the Japanese to discuss and that is the islands issue," he said. "If we agree to discuss this issue, it would be tantamount to agreeing on the existence of a sovereignty issue."

Tokyo's purchase of three of the uninhabited islands spurred major Chinese protests and a boycott of Japanese goods in China.

The United States, which has a hefty military presence in Japan including on the southern island of Okinawa, close to the disputed isles, has expressed concern about the dispute and has been keen to see a diplomatic solution.

(Reporting by Randy Fabi in NUSA DUA, Indonesia, and Ben Blanchard in BEIJING; Editing by Paul Tait and Robert Birsel)