Washington was deeply concerned about the "potential for further violence" after Nkurunziza returned to the country, said Jeff Rathke, a State Department spokesman.
General Godefroid Niyombare, who launched the coup in the central African nation earlier in the week, told AFP by telephone that he wanted to give himself up.
Coup leaders have been arrested and others have been forced on the run by loyalist troops.
The dramatic end to the coup attempt ended 48 hours of uncertainty over who controlled the small, landlocked and impoverished nation, which has been gripped by a political crisis over Nkurunziza's controversial bid to run again for the presidency.
The president's decision to run "has and will continue to exacerbate instability... In the country," Rathke warned, only a day after he had stressed Washington still regarded Nkurunziza as the legitimate president.
"This threatens the viability of the Burundian government, and it increases the risk of violence and instability that can threaten donor support."
Rathke renewed calls on all sides to exercise restraint, after the State Department late Thursday ordered the families of all US government staff to leave the country.
Amid continuing uncertainty, the US embassy was closed on Friday.
Washington could slap visa bans on anyone seen as responsible for such violence.
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