However, Beijing again demanded that he correct his views on the country's role in the second world war.
Mr Abe said Japan had not made any direct overtures, but he hopes the leaders can meet to help resolve antagonism over territorial disputes and historical issues.
He said: "At the moment, there is no plan for a summit meeting, but since there are some difficulties and issues we should be speaking together without setting any preconditions.
"I would really like to explain the intent of my visits to the Yasukuni shrine directly to them.
The door to dialogue is open. I would like to hold Japan-China and Japan-South Korea summit meetings."
Japan colonised Korea and occupied parts of China before and during the war and that often brutal legacy still taints relations with its neighbours decades on.
China and South Korea reacted angrily to Mr Abe's Boxing Day visit to the shrine in Tokyo.
The Chinese Foreign Ministry has said Mr Abe will not be welcome in Beijing until he admits that the trip was a mistake.
The ministry said that to improve relations with Beijing, Mr Abe "needs to correctly view and deeply reflect on the Japan's militarist history of external invasion and colonialism, show sincerity and make concrete efforts to improve ties with neighbouring countries".
The Chinese also accused Mr Abe of hypocrisy, while South Korean president Park Geun-hye blamed Tokyo for the strained ties and said any summit with Abe could be held only after sufficient preparations.