Dmitry Medvedev's address to the Council on Foreign Relations on Saturday followed a summit of the Group of 20 rich and emerging economies focused on the financial crisis.
"We welcome the election of Barack Obama, and count on him to take steps to overcome the problems that have built up in Russian-American relations," he said.
"I believe that currently, Russian-U.S. relations lack the trust that they need."
Medvedev's speech came in sharp contrast to his state of the nation address, a day after Obama's election victory, when the Russian leader pledged to deploy tactical missiles near Poland in response to U.S. missile shield plans.
On the missile shield issue, Medvedev said on Saturday: "Our actions will be only a response, and will only come if the [missile shield] program continues in a manner that is unacceptable to us... This issue is not closed, I personally am ready to discuss it, and I hope that the new president and the new administration will have the will to discuss it. The signals that we have received so far suggest that our new partners are thinking about this problem," he said.
Following Medvedev's speech, former secretary of state Madeleine Albright, who has been acting as an informal adviser to Obama, said the Russian position provides "a huge opportunity" in bilateral relations.
"I don't think Russia and the United States have ever had two such young and impressive presidents who will be able to communicate in a new generational way, and I as the old lady here would like to wish you all the best of luck," she said.
Washington recently said it had provided new proposals to ease Russia's concerns over the planned deployment of 10 U.S. interceptor missiles in Poland and a tracking radar in the Czech Republic, which the Bush administration has said are needed to counter possible attacks from "rogue" states such as Iran.
After Obama's presidential election victory, one of his senior foreign policy advisers, Denis McDonough, said the president-elect was not committed to the missile shield, and would only continue with the project if its effectiveness was proven.