A court spokeswoman told reporters in Moscow that Klein had 10 days in which to lodge an appeal with Russia's Supreme Court.
If he did not do so, the extradition would take effect, she added.
Klein is also alleged to have worked as a mercenary in Africa and spent time in jail in Sierra Leone, AFP reported.
Colombia's paramilitary groups were ostensibly organised as private armies in the 1980s to combat fighters from groups such as the Revolutionary Armed Forces of Colombia (Farc).
However more than 3,000 people died in fighting involving the paramilitary groups, including four candidates in Colombia's 1990 presidential election.
Many paramilitary groups were decommissioned following a series of peace deals with the Colombian government.