Ryan Lochte has reportedly been offered a deal by Brazilian prosecutors in which charges would be dropped in return for a BRL R$70,000 (£16,000/$20,000/€19,000) payment.
According to TMZ Sports, the deal has not yet been accepted.
Lochte was among four United States swimmers that initially claimed they had been robbed at gunpoint when returning to the Olympic Village during Rio 2016 in August, before closed circuit television footage emerged of the men vandalising a petrol station following a night out.
The incident provoked outrage in Brazil and negative reactions across the world.
Lochte, who was seen as the instigator of the confrontation, has already been suspended for 10 months by USA Swimming.
The six-time Olympic champion has been charged in Brazil with "falsely reporting a crime".
A request for his case to be dropped was rejected by a Brazilian judge in October.
"We don't comment on a pending case, but are hopeful it will be resolved to the benefit of all parties, including the people of Brazil, the Brazilian Government and Ryan," Lochte's lawyer Jeff Ostrow told TMZ Sports in relation to the alleged payment offer.
Lochte has been accused of having pulled a metal advertisement in a frame to the ground and it is then claimed he "stood up and began to yell at guards" after they were caught.
He has admitted he "over-exaggerated" what had happened, although he has continued to claim he was not telling an outright lie.
The penalty for falsely filing a crime report carries a maximum penalty of 18 months in prison under Brazilian law and Lochte can be tried in his absence.
The payment offer was seen as an alternative to such a trial.
A fee of BRL R$35,000 (£8,000/$10,000/€9,500) was paid by James Feigen, one of the other swimmers involved, before he left Brazil in August.
The two others involved, Gunnar Bentz and Jack Conger, were initially prevented from flying out of Brazil until they had been questioned by police.
Bentz, Conger and Feigen have each been given four month suspensions by USA Swimming.