Burns was quoted by Radio Voice of America as saying that the US supported the EU's policy towards Croatia, which also wants to see Gotovina in The Hague before opening membership talks.
Burns was speaking to reporters in Washington following his Balkan tour, which included visits to Sarajevo, Belgrade and Pristina.
He said that Gotovina was the third most wanted indictee after Bosnian Serb wartime leader Radovan Karadzic and his military commander Ratko Mladic.
Burns believes that Mladic's days at liberty are numbered and hopes that he will surrender or be arrested by the 10th anniversary of the Srebrenica massacre on July 11.
Mladic ordered the massacre of 8,000 men from Srebrenica, which is the worst single crime in Europe since the Second World War. For the United States the arrest of war criminals is a priority, Burns said.
Karadzic and Mladic must be arrested and until that happens neither Belgrade nor Banja Luka will have normal relations with the US administration, he added.
Burns said he believed authorities in Belgrade would do all in their power to arrest Mladic in view of the fact that the present government had done more in a few months than the previous government had in ten years to locate and arrest war crimes indictees.
He strongly criticised Bosnian Serb authorities for delaying reforms and rejecting responsibility for atrocities committed during the 1992-1995 war in Bosnia-Herzegovina.
I feel sincerity on the part of Bosnian Muslim and Croat authorities and relatively feeble sincerity on the part of the Bosnian Serb leadership. I met Bosnian Serb Foreign Minister Mladen Ivanic and had a feeling he was trying to evade most of my questions, Burns said.
Burns described the Bosnian Serbs as a community that does not have a good reputation in multiethnicity.