The arrest followed a decision earlier this month by U.S. authorities to hand over a file on Kovacevic's case to the Serbian judiciary, Tomo Zoric, a spokesman for Serbia's prosecution office, told The Associated Press.
Serbia has refused to extradite Kovacevic to the United States, saying its laws do not allow it. The case has strained relations between the two countries.
U.S. Senator Charles Schumer, a leading Democrat from New York, thanked Serbia's government for their actions - but made clear Kovacevic should be extradited.
"It is far better to have Kovacevic detained than on the basketball court and now we must work to ensure that he is returned to the United States as soon as possible to face justice," Schumer said in a statement.
"This arrest should be a wake-up call for Mr. Kovacevic and for anybody complicit in his flight from justice," said U.S. Senator Hillary Rodham Clinton of New York. "While Mr. Kovacevic's arrest is a step in the right direction we must continue to work together to ensure that he returns to the U.S. to face prosecution."
It was not immediately clear if or when Kovacevic, 21, would be formally charged in Serbia. An investigative judge ordered him to remain in prison for at least 30 days pending a formal investigation.
Kovacevic's lawyer, Borivoje Borovic, said Kovacevic used silence as his defense during the judge's questioning Tuesday. Kovacevic's father, Petar, told the AP that his son "voluntarily went to a police station for questioning."
Serbia's police media office said that a former Serbian consul in New York also was arrested Tuesday. The consul allegedly helped Kovacevic by providing emergency travel documents that helped the 21-year-old flee the U.S. last June.
Both the ex-consul and Kovacevic are suspected of forging the travel documents, Zoric said. The former consul, Slobodan Nenadovic, was also ordered to remain in jail for 30 days pending the formal inquiry.
Kovacevic, who played basketball for Binghamton University in upstate New York, jumped bail after being charged with assault. Steinhauer, 22, only recently emerged from a coma.
Kovacevic has signed a contract to play for a basketball team in a regional Serbian league.
