The court rejected an appeal by Klein's defense, which said the defendant was persecuted for political motives and that the five-year time limit for the sentence to be enforced has expired. Klein has said he may face torture in Colombia.
"Colombia has been plunged into chaos... there is neither an opportunity nor a desire to protect me there. Extradition to Colombia would mean a death sentence," Klein said.
Klein also claimed that organs of murdered prisoners are freely traded in Colombia.
Colombia had asked Russian authorities to detain and extradite Klein to enforce a sentence for assisting terrorists handed down by a Colombian court in 2001.
Colombia's ambassador to Russia and Belarus, Diego J. Tobon, earlier said Klein had repeatedly made statements in the media admitting to having visited Colombia since 1989 to train people involved in illegal drug trafficking and terrorism.
The ambassador said there was no political motivation in this case and that it was considered to be in strict compliance with the law.
Lawyers say Russia and Colombia have an understanding on extraditions of foreign nationals between the countries.
Prosecutor Anatoly Bashmakov told the Russian Supreme Court that in 1989-1990 Klein, pretending to work for a security guard company, trained terrorists who later used their skills to carry out a series of attacks in the country. He also said Colombia had guaranteed that no inhumane treatment would be used when Klein returns.
