The two men will try to convince Mr Milosevic that the time has come for him to tell his associates in Bosnia that their moment of maximum gain has passed and that an inevitable process of international condemnation and military assault awaits the Serbs if they block a peace agreement.
'There is no doubt he is a key player and that he is in a position to push this peace process home to the conclusion that Vance and Owen are trying to bring it too,' said Fred Eckhard, spokesman for the two negotiators.
The two men left for former Yugoslavia after talks adjourned in Geneva between the Muslim-led government of Bosnia and leaders of its Serbian and Croatian minorities. They have been summoned to another session next Sunday, at which Lord Owen and Mr Vance hope they will sign agreements to end nine months of armed conflict in Bosnia and to rebuild the country around 10 autonomous provinces.
The Bosnian Serb delegation has so far refused to sign because the provisions for autonomy prevent the creation of the state within a state which they desire.
'This will be top of the agenda in Belgrade,' Mr Eckhard said. The draft agreement says: 'The provinces shall not have any international legal personality and may not enter into agreements with foreign states.'
The Bosnian Serb leader, Radovan Karadzic, has declared, 'I do not know of any Serb who would accept anything less than a state.' However, it was made known in Geneva that Mr Karadzic will be on hand during the Belgrade meeting today and may join the discussions. Diplomats close to the negotiations said the aim would be to persuade both President Milosevic and Mr Karadzic, who they see as his creature, that the moment of compromise has arrived.
There are signs that the tide of battle has slowly turned against the Serbian forces in Bosnia. The government in Sarajevo, believing either military intervention by the West or dramatic new support from the Islamic nations to be at hand, has discovered a new confidence in the last few days. President Alija Izetbegovic and his delegation deftly turned the talks away from Muslim reservations about the Vance-Owen plan to focus on the issue of sovereignty and the resistance of Mr Karadzic.
Mr Eckhard left no doubt that the two negotiators intended to push all the parties as hard as they could for a signed set of agreements at the next talks. If the talks break down because one or other side refuses to sign, Mr Eckhard said, Lord Owen and Mr Vance would hand the issue back to the UN Secretary-General and the Security Council. 'They will not hesitate to identify the party that stood in the way of an agreement,' he said.
They would be urging the maximum military restraint during the Belgrade talks, he added, noting that both men were very concerned at rising tensions in the autonomous Serbian republic of Kosovo, inhabited mainly by Albanian Muslims.
The twin issues of Bosnia and Kosovo - interlinked by the urge to restore calm in the first and avert fresh conflict in the second - are acting as a single spur to what may prove to be the last great effort by Lord Owen and Mr Vance to impose a diplomatic peace in the Balkans.