"I do not rule out that, under the new president of the USA, Russian-American relations could significantly improve," Gorbachev told Russia's Interfax news agency.
"It is now necessary to start bilateral dialogue at the highest level... And to prevent failure of the dialogue under any circumstances," he said, adding that "a lot also depends on Russia's position".
Russian President Vladimir Putin - who was one of the first world leaders to congratulate Trump on his electoral win -- said Moscow was ready to contribute to rebuilding relations with the United States with Trump as president.
During the campaign, Trump has said emphatically that he is willing to work with Putin, even praising him as a superior leader to outgoing US President Barack Obama.