Diplomatic relations between the two countries plunged to a post-Cold War low after Russia took military action in August to crush neighbouring Georgia's attempt to regain contol over South Ossetia.
The region - which broke away from Tbilisi's control in the 1990s - has been recognised as an independent state by Moscow, but much of the West has criticised Russia for its support of the separatists.
'Gesture of goodwill'
In what has been interpreted as a gesture of goodwill, Medvedev signalled his country would be willing to compromise over American plans to deploy a radar system in the Czech Republic and an anti-missile barrage in Poland.
Medvedev also indicated Russia would hold off on a possible military response to the US project. He said Moscow would not be the first to escalate the situation.
"We will not do anything until America takes the first step," he said.
Russia views the project as a threat to its own security.
"It's better to have a global missile defence rather than kind of fractured national elements"
Dmitry Medvedev, Russian president
Just one day after Obama's election victory, Medvedev announced plans to deploy missiles in Kaliningrad, a Russian enclave which borders Nato military alliance members Poland and Lithuania.
However, the Kremlin hopes the Obama administration will not pursue the missile defence shield project with the same vigour as George Bush, the outgoing US president.
"[The] first signal we received shows that our partners think about this programme rather than plan to simply rubber-stamp it," Medvedev said.
Joint defence plan?
For the first time, Moscow signalled it might accept something less than a total rejection of the missile defence system project by the US. The Russian president suggested it might be possible to agree a joint anti-missile system.
"It's better to have a global missile defence rather than kind of fractured national elements," Medvedev said.
"We have a chance to solve the problem through either agreeing on a global system or, as a minimum, to find a solution on the exisiting programmes which would suit the Russian Federation."
In a marked shift in rhetoric, Medvedev insisted there was "no anti-Americanism in Russia". Instead, there were "problems which have been piling up and chances to solve them".
The Russian president also dismissed suggestions that he made anti-US comments on the day after Obama's election win in a bid to put pressure on the president-elect.
"You think it was blackmail,"he said. "With all respect to the USA, I did not even think then what day it was."
Medvedev said he would meet Obama soon after he takes office in January.