Trapattoni was already under significant pressure prior to the Republic of Ireland's 1-0 defeat against Austria in Vienna on Tuesday, which was settled by a late strike from Bayern Munich's David Alaba.
Ireland now sit six points behind second-place Sweden in Group C with two games to go, have an inferior goal difference to their rivals and face a trip to leaders Germany next month.
And while Trapattoni has made it clear that he will not be stepping down as, the FAI plan to review his future this week.
An FAI statement read: "The recent results against Sweden and Austria were very disappointing for everyone involved in Irish football, the manager, the players and supporters.
"Our aim of qualification, although mathematically possible, is now highly unlikely even though the team will always do everything within their power to keep that hope alive.
"The FAI Board of Management will now consider the current position over the coming days."
Trapattoni, who took charge of Ireland in 2008, told RTE following the Austria defeat: "For me it's simple. I continue 100 per cent with the commitment.
"Obviously the FAI must now take the decision. I said (on Monday) that if they give me the opportunity to continue for the next campaign I'm sure this team can increase very much.
"But it will be for the other manager, maybe. It depends what they decide. I wait for the decision. Maybe they are ready for another manager, I don't know."
Trapattoni felt Ireland were unfortunate to come away from the Ernst Happel Stadion empty-handed.
"I think we played well in the first half, very well," he added.
"In the second half we lose a bit of strength. Third position for us was the aim. It's a pity."
Ireland midfielder Paul Green cut a dejected figure when delivering his assessment of the game.
"Mathematically we can still qualify but it’s not looking the best to be honest," he said.
"The game against Sweden (which Ireland lost 2-1 on Friday) was a massive result for us. Obviously it didn't go our way and we just wanted to put on a show here. But we didn't take our chances and a bit of quality from Alaba finished us off.
"We've given everything and it's a bit of a sucker punch the lad scoring near the end."
