Asked if Japan can address Putin's concern about the possibility of USA forces building a base on any of the islands if they are handed over to Japan, Abe said his government has "not yet" gained the consent of the United States to keep them free of a US military presence.
The Kremlin published a summary of Mr Putin's "greeting message" to Mr Trump on Sunday.
"[Putin] confirmed that Russian Federation is open for dialogue with the United States of America on the most wide-ranging agenda", the statement said.
The message to Trump was among dozens of holiday greetings Putin sent to other world leaders, each tailored to reflect a bilateral theme.
In a dispute over a separate, longer-range missile system, Washington has threatened to pull out of the landmark 1987 Intermediate-Range Nuclear Forces (INF) Treaty, alleging that a new Russian missile, the Novator 9M729 (called SSC-8 by Nato), violates the pact.
Since then, the Kremlin has repeatedly said it is open to dialogue.
Putin wished "well-being and prosperity to the British people", the Kremlin said.
Russian Foreign Minister Sergei Lavrov was quoted as saying that now it was up to the United States whether to hold a new meeting in 2019.
Russian Federation has been flexing its military muscles overseas in recent years, and its involvement in the Syrian conflict and Ukraine has soured relations with the West.
Russian Federation and the U.S. have never seen eye-to-eye on several issues.
Deadly blast partly destroys Russian apartment block
Officials warned that two more sections of the Soviet-era high-rise on Karl Marx Street were in danger of collapsing. The blast rocked the apartment complex at around 4 a.m. local time, completely collapsing one of its sections.