At the United Nations International Criminal Tribunal for the Former Yugoslavia, to use the official title, the presiding Judge Jorda is expected to issue, for only the second time, warrants for their arrest.
After a good deal of bravado, the move - conferred by Rule 61 of the tribunal - is to step up the pressure on the international community.
Rule 61, otherwise known as the "voice of the victims", operates where, as in the case of Mr Karadzic and General Mladic, original warrants of arrest from the tribunal have not been executed. They were first indicted for genocide after the fall of Srebrenica a year ago.
While the lack of active co-operation by the Serbian President Slobodan Milosevic means that the indictments have never been served, the rule allows for hearings in open court, which ended on Monday, of the evidence on which the indictment was originally founded.
Under the rule, if judges consider there are reasonable grounds for believing the charges, they endorse the indictment and issue an international warrant.
The difference between the two warrants is that the second reconfirms the indictments, while putting an obligation on the rest of the world to help bring them to justice. All UN member states will now be obliged to comply.
The tribunal, the first international war crimes court since Nuremberg, has power to declare that there has been a lack or refusal of co-operation by the authorities who were supposed to serve the indictment on the accused. Official notification to the Security Council would normally follow.
Because the Rule 61 procedure is not a trial of the accused in absentia - under the tribunal's statute every accused has the right to attend his trial - the court cannot determine guilt. The idea is to help the tribunal resist justice being defeated simply by the non- appearance of the accused.
Once the international warrants have been issued, responsibility for bringing in Mr Karadzic and General Mladic will no longer rest with the Serbian authorities, but becomes a clear obligation on all states.
The final upshot is not so easy to gauge. Admiral Leighton Smith, the US commander of Nato forces in Bosnia, has said he was ready to make the arrests, but was awaiting orders from Nato.
Like any other kind of international court with no international police force specifically charged with implementing its orders, its effectiveness will inevitably be conditioned by the political impact it can muster.
On this score alone, things will become harder for Mr Karadzic and General Mladic from today. They will have been publicly branded as international fugitives while the Bosnian Serb territory in which they shelter becomes a kind of open-air jail. Stepping outside it could risk immediate extradition or transfer to The Hague.