Giovanni Trapattoni's side will meet the Omanis for the first time at Craven Cottage on Tuesday, September 11.
The game will come four days after Ireland play their opening World Cup qualifier in Kazakhstan.
It will be the third time the Republic have played at Fulham, familiar territory for defender Stephen Kelly and midfielder Damien Duff, under Trapattoni.
Kelly said: "Everyone knows the Republic has the best supporters in the world, so we are expecting there to be a great atmosphere and are looking forward to welcoming Oman and their fans to Fulham's home ground."
For Trapattoni, the clash with Oman - managed by former Rangers boss Paul Le Guen - will give himself and his staff a further opportunity to hone their plans for the World Cup campaign.
The Italian said: "I am looking forward to returning to Craven Cottage for this match, which will be a useful opportunity to prepare further for our World Cup qualification campaign."