A White House spokeswoman emphasised on Wednesday that "US ability to assist Yugoslavia depends on Belgrade's relations with the tribunal".
The two presidents met on Wednesday in Washington, where the Yugoslav leader also met Secretary of State Colin Powell and other leading administration figures.
Mr Kostunica said that his country was drafting a law to make it possible to co-operate with the UN tribunal in The Hague, but refused to name a date for Mr Milosevic to be handed over.
"Nothing is easy in this situation", Mr Kostunica said.
He has maintained that Mr Milosevic, who is in prison in Yugoslavia, must be tried at home for corruption and abuse of power before anyone else has a chance to prosecute him.
UN indictment
The UN tribunal has indicted Mr Milosevic for crimes allegedly committed in 1999 by Yugoslav forces in Kosovo.
Prosecutor Carla del Ponte has demanded that Belgrade hand Mr Milosevic and other suspects over for trial in The Hague without delay.
A White House spokeswoman emphasised on Wednesday that "US ability to assist Yugoslavia depends on Belgrade's relations with the tribunal".
Mr Powell approved the release of $100m in US aid to Yugoslavia after Mr Milosevic was arrested on 1 April.
But he has reserved support for a planned donors' conference to be held in June.
Mr Kostunica is the first democratically elected Balkan leader to be invited to the Bush White House, a sign of how much emphasis the administration puts on Yugoslavia and Mr Milosevic.
Mr Bush spent about 10 minutes with Mr Kostunica, joining a meeting already in progress with Vice-President Dick Cheney and National Security Adviser Condoleezza Rice.