Although Russian officials denied that Medvedev made a specific proposal, he moved to broach the issue of uniting a missile shield being built by the 28 NATO allies with Russia's own version during a closed-door meeting with NATO leaders in Lisbon over the weekend.
"I was present in the room. ... Our president made no concrete proposal," Russian Ambassador to NATO Dmitry Rogozin was quoted saying by the Defense News Web site.
"He simply expressed the principles of cooperation that we support, such as equality between Russia and its NATO partners," the envoy said.
The Wall Street Journal, however, reported that Medvedev had proposed that Russia take "responsibility for shooting down missiles fired towards Europe and which fly over its zone of responsibility."
NATO leaders said the issue should be probed by technical experts who will examine ways the two systems could cooperate. A follow-up on the issue was expected in June when defense ministers from Russia and NATO member countries are set to meet in Brussels.
Washington's plan to host a long-range missile defense shield in the European continent is intended to protect the United States and Europe from feared Iranian attacks. That plan was developed and negotiated by the previous U.S. administration and it angered Russia. It has since been reconfigured by U.S. President Barack Obama.
Military analysts said a cooperation with Moscow would mean that NATO's system sensors would not be pointed toward Russia.
"Obama simply said that matters needed to be made clear, that we are at the beginning of the road and that he is ready to discuss everything. We are completely in agreement about working together to neutralize the risk coming from regions south of Europe," Rogozin was quoted saying by Defense News.
Russian officials have said that Moscow's proposal goes beyond missile cooperation, but an exchange of information and alert procedures.
Russian military officials have expressed doubt about Medvedev's proposal.
Nikolai Makarov, chief of Russia's General Staff, said in the official Rossiiskaya Gazeta newspaper that the idea of Russia shooting down missiles headed over its territory for Europe and vice versa was simplistic.
"For military men on both sides, this supposition looks, to put it gently, far-fetched. The fact is that any country with missile-defense systems would shoot down missiles approaching its borders without any international agreements
Moscow and NATO severed contacts two years ago after Russia was accused of invading its tiny NATO-aspiring neighbor Georgia. The recent summit was aimed at restoring those relations.