Republic of Ireland boss Martin O'Neill labelled Ronald Koeman a "master tactician of the blame game" as their row over James McCarthy's fitness escalated on Friday.
McCarthy, 26, missed Everton's past two matches but linked up with the Republic squad in the international break.
He suffered a recurrence of a hamstring injury, ruling him out of Saturday's derby at Liverpool.
Toffees boss Koeman had accused O'Neill of "not protecting" the midfielder.
In his news conference on Friday, a clearly angered Koeman read out a prepared statement in which he added: "Clearly James was not fit to play. We advised extreme caution. He was not fit to play.
"In my opinion, the Ireland manager in this instance was not protecting the player. I am not surprised but I am disappointed."
'Quiet introspection may serve the Everton manager'
In response, O'Neill issued his own statement later on Friday.
It read: "Once again the Everton manager, master tactician of the blame game, has struck out in his comments today, criticising both myself and James McCarthy.
"Perhaps a review of Everton's pre-season programme might provide some enlightenment.
"James had a magnificent tournament for the Republic of Ireland last summer during at Euro 2016, playing his last game in very late June.
"He then returned to Everton after a very short break, but only 11 days later, he played his first of three games, all within an eight-day period, against Real Betis, Manchester United and Espanyol. Overloading?
"It should be added that James last played for his country on 9 October - almost half a year ago. Since that time he has been totally under Everton's supervision.
"James is diligent and conscientious in his professional preparation. Perhaps, in this instance, quiet introspection may serve the Everton manager and his medical staff better."McCarthy was injured in the warm-up before the World Cup qualifier against Wales
McCarthy was named in O'Neill's starting line-up for the 0-0 draw against Wales but pulled out of the team after suffering an injury in the warm-up.
In October, Koeman said McCarthy, who has been struggling with a hamstring injury for several months, had been "massively overloaded" by the Republic.
That came in response to O'Neill's claim that the Dutchman was "bleating" about the matter.
Former Republic midfielder Roy Keane, O'Neill's assistant, then said Everton players must "toughen up".
'This is not the first time' - Koeman's full statement
"James had an injury when he reported for Ireland duty last week and our medical team made the Ireland medical team aware of this.
"Everton's medical team advised extreme caution, not only due to the current injury but also due to previous injuries.
"The assessment by the Ireland medical was that it would be a high risk for James to play against Wales.
"But of course James has a strong desire to play for his country so when asked if he was fit to play he said he felt he was fit. And he was selected to start the game but withdrawn.
"In my opinion, in this instance it was not protecting the player. James was clearly not fit to play. He only trained for two days with Ireland and broke down in the warm-up.
"He has only played only one game from the start in 2017 and he did not play for three weeks before the Wales game. In my opinion he would need at least one full week of training sessions to be declared fit.
"I spoke to James, he needs to take responsibility for this. But this is not the first time."'Don't underestimate power of Anfield' - Bosses on Merseyside derby