Serdyukov is on a two-day official visit to NATO member Turkey.
"I believe that we need to resume dialogue and search for constructive solutions in all areas of cooperation on strategic stability," the minister told a news conference after talks with his Turkish counterpart, Vecdi Gonul.
Russia's envoy to NATO Dmitry Rogozin said on Tuesday that ambassador-level consultations between Russia and NATO could take place before the end of the year.
In an indication that dialogue is possible, an alliance spokesman said earlier on Tuesday that the foreign ministers of 26 NATO member countries may decide to resume the work of the Russia-NATO Council at their meeting in early December.
However, NATO Secretary General Jaap de Hoop Scheffer said the alliance would support efforts by former communist nations to join the military alliance regardless of opposition from Russia.
He added that NATO was not prepared to sacrifice the alliance's enlargement for good relations with Russia.
"NATO will continue to work with all countries that aspire to join our alliance," he said. "Do we have to choose between good relations with Russia and further enlargement? My answer is no - we will not choose, will not sacrifice one for the other."
Scheffer said he would like to like to meet with Russian officials to straighten out their differences.
"I am ready to go to Moscow to talk to the Russian leadership. I am ready for it, I think NATO allies are ready for it. I hope I am welcomed there soon," he said.
He also backed calls for an independent inquiry into the Russia-Georgia conflict, while condemning Russia's recognition of Georgia's separatist provinces of South Ossetia and Abkhazia, and saying Moscow had used "disproportionate force."
"I support those that want an independent inquiry into what happened exactly. But independently of who fired the first shots, the use of force by Russia was disproportionate and the subsequent recognition of parts of Georgian territory cannot in any possibility be considered legal," he said.
