Ties between Russia and Britain have plunged to a post-Cold War low since the murder of the former Russian security officer and London-based Kremlin critic Alexander Litvinenko in 2006 and Russia's refusal to extradite London's chief suspect.
"We are open to the re-establishment of co-operation to the full extent," Medvedev told the paper. But he added "time would tell" whether progress was possible when he meets British Prime Minister Gordon Brown for the first time - probably at July's Group of Eight summit in Japan.
The U.K. introduced visa and cooperation restrictions for Russian officials in July last year in the wake of the extradition dispute, with Moscow reciprocating. Bilateral ties were further strained by the closure of British Council offices in St. Petersburg and Yekaterinburg for alleged tax and status violations early this year.
Medvedev told the newspaper the closure of the British cultural organization was "not very surprising because these types of organizations are traditionally used for the collection of information."
The president-elect dismissed suggestions that this week's raid on TNK-BP, the Russian-British petroleum group, and industrial espionage charges against one of its employees, and a member of the British Alumni Club with links to the British Council and embassy, were attempts to sabotage efforts to improve ties.
"I would like the final decision to be taken by a court and not by the judgment of analysts or politicians," Medvedev told the daily.
The handpicked successor to President Vladimir Putin, Medvedev, 42, is to be sworn in on May 7. The popular Putin is expected to be prime minister.