Kovacevic fled to his native Serbia in early June after he was freed on $100,000 bail. That touched off a diplomatic crisis between Serbia and U.S. officials demanding his return.
"Representatives from the U.S. Embassy and the Ministry of Foreign Affairs of Serbia met jointly with the parents of Miladin Kovacevic and their attorney on July 6th at the Ministry of Foreign Affairs building in Belgrade," a U.S. Embassy statement said Monday.
"The purpose of the meeting was to discuss the current situation regarding the flight of Miladin Kovacevic, who has been charged with assault in the State of New York and to answer questions from the family regarding the U.S. judicial process and the ramifications for Mr. Kovacevic of his flight from the U.S.," the statement said.
"The United States has not formally requested extradition at this time," it added. "We sincerely believe that it is in the best interests of Mr. Kovacevic to return voluntarily to the U.S., rather than to remain a fugitive. The meeting was constructive and the Kovacevic family and counsel were receptive to the points raised in the discussion."
Kovacevic's lawyer Veselin Cerovic said last week that he fled because he doesn't trust the U.S. justice system.
U.S. police say Kovacevic was at a bar with friends when Steinhauer danced with one of their girlfriends, sparking the dispute.
Kovacevic's parents claimed their son is innocent and that Steinhauer started the fight in a drunken state.
"He did not hit or kick Bryan Steinhauer," father Petar Kovacevic told reporters in Belgrade, adding that his son just separated others who were involved in the brawl.
Kovacevic's mother Branka, who was in the U.S. when he fled, said that they decided to leave after receiving unspecified "telephone threats." She said that her son is currently "depressed" and that it was not clear when he would return to normal life.
New York congressmen have asked U.S. Secretary of State Condoleezza Rice to pressure Serbia to return him.
Last week, the U.S. government asked the Serbian Foreign Ministry to send Kovacevic back to America for trial, but Serbian officials have said they want to see the case against him so they can consider whether he should be tried in a Serb court instead.
Serbia's laws do not allow the extradition of its citizens to the United States. Two Serbian consulate employees in New York were accused of giving Kovacevic emergency travel documents that helped him leave the country.
Meanwhile, two Bosnian nationals accused of taking part in the fight together with Kovacevic will stay in jail for the time being.
Broome County Court Judge Martin Smith wants more information before deciding whether to set bail for Sanel Softic, 21, and Edin Dzubur, 24.
They face felony counts of assault and gang assault in connection with the May 4 attack on Steinhauer, who remains in a coma.
Lawyers told the judge Monday that Softic and Dzubur could wear electronic monitoring devices if they are released.
