The League One outfit have been drawn against Premier League side Everton in the third round of the competition next month.
It was revealed on Friday that the match at Goodison Park is due to be played on Thursday, 9 January with a kick-off time of 7:45pm.
However, Peterborough have expressed their dissatisfaction with the timing and a lack of consideration for their supporters.
"We were very confident of taking a full allocation of tickets for our ever first visit to Goodison Park and what is the final ever third round tie there," a statement read.
"On learning that the tie will take place on Thursday 9th January 2025 we are incredibly disappointed that the needs of our supporters have not been taken into consideration in this decision.
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"Although we have an appreciation that Liverpool and Everton are unable to play on the same day and with Liverpool also drawn at home it was unlikely the game would take place on Saturday, we feel a better solution could have been found to ensure all our supporters who wish to travel to the game would have been able to do so.
"Following confirmation from the FA of the decision to play the fixture on the Thursday, we contacted them to ask for the decision to be reversed based on the following criteria:
"1. There are no viable trains from Liverpool to Peterborough scheduled on the evening of the match meaning supporters must travel by car or coach.
"2. Supporters will be required to take two days off work to attend this incredibly important fixture and our younger fans will not be able to attend due to school commitments.
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"3. We have never played a competitive game at Goodison Park and interest in the game is very high.
"4. An expected reduced home attendance would reduce the potential revenue for both Peterborough United and Everton."
Peterborough CEO Dawn Gore also said: "Like our supporters, when the draw was made on Monday night, we were all very excited about travelling to Goodison Park.
"We were inundated with supporters requesting details for ticketing within hours of the draw being concluded. To learn that the game has been arranged for a Thursday night is bitterly disappointing.
"Whilst we understand there are logistical challenges, we do not feel the supporters have been considered in this approach.
"We have made our feelings very clear to the decision-makers at the top of the game and want to assure our supporters that we are just as disappointed as they are about when this game will take place and from a financial standpoint, the Club are set to lose out on revenue that would have been greatly appreciated as a club in Sky Bet League One."
Peterborough beat Newport County and Notts County in the first and second rounds of the FA Cup.
The Posh, who are 15th in League One, exited at the third round last season when they were defeat 3-0 by Leeds United.
Aston Villa host West Ham on Friday, January 10 in the first televised match of this season's FA Cup third round on ITV.
Liverpool vs Accrington Stanley is the lunchtime kick-off on Saturday before Manchester City face Salford City at home in the evening.
Tamworth, the lowest-ranked side left in the competition, will play against Tottenham on Sunday, with fellow National League side Dagenham & Redbridge travelling to Millwall on the Monday.
Arguably the biggest tie of the round sees Arsenal face Manchester United on Sunday afternoon at the Emirates Stadium.