The Republic of Ireland may take part in a consolation tournament featuring Italy and Holland in the USA next summer.
The United States Soccer Federation is hoping to attract high-profile teams that failed to qualify for the World Cup, including Chile and Wales, the Irish Daily Mail reports.
It is understood that FIFA is not opposed to the event as long as it does not clash with the Russia World Cup, which begins on June 14.
Other teams in the mix are Nothern Ireland, Scotland, Ghana, Cameroon and Ivory Coast, along with the hosts.
The FAI does not have any confirmed fixtures for next summer and are understood to be considering the possibility of returning to the States, where Ireland toured in 1992, 1996, 2000 and 2014.
Meanwhile, Burnley boss Sean Dyche has welcomed back the Irish contingent of Robbie Brady, Jon Walters, Jeff Hendrick, Stephen Ward and Kevin Long, fresh from play-off heartbreak against Denmark, for Premier League duty.
"They are good but they have taken it as a bit of a knock, not just the fact that they haven't made it to the [World Cup] but the game [as well]," he said.
"But they are in a group here who accept them back very quickly and say 'C'mon lads, we've got to get on with it'."
