CCTV footage later contradicted their story, showing the swimmers had vandalised a petrol station.
Mr Lochte left Brazil on Tuesday but Mr Feigen stayed behind and was detained.
The two other members of the group, Gunnar Bentz and Jack Conger, were pulled off a flight in Rio on Thursday night and questioned by police.
They deny any involvement in the false robbery claim and were allowed to leave for the US on a later flight.
Breno Melaragno, Feigen's lawyer, said: "After a long deliberation, this agreement was reached… he will donate 35,000 reais ($10,800) to an institute, and with that the case is resolved.
"After this donation is done, his passport will be given back to him, and he will be free to return home."
Mr Melaragno did not say which charity, or institute, would receive the donation.
The bizarre story began when Mr Lochte told a reporter that the four men were robbed at gunpoint by thieves disguised as police officers.
In a colourful description of the fabricated robbery on 14 August, the swimming star said he had a gun pressed to his forehead but refused to get on the ground.
But when police began to investigate, they could not find any evidence of the crime. Rio's civilian police chief Fernando Veloso has now ruled out any possibility that it occurred.
"No robbery was committed against these athletes. They were not victims of the crimes they claimed," Mr Veloso said.
He told reporters that one or more of the athletes had instead vandalised a toilet in a petrol station and then offered to pay for the damage.