Koeman and O'Neill have been regularly rowing this season over the fitness of midfielder James McCarthy.
The latest incident came when McCarthy was picked by O'Neill to face Wales, only to pull out with injury just before kick-off.
Speaking at his press conference ahead of the Merseyside derby Koeman said "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."
Later on Friday evening O'Neill hit back and he didn't mince his words.
"Once again the Everton manager, master tactician of the blame game, has struck out in his comments today, criticising both myself and James McCarthy," O'Neill said in statement released by the Irish FA.
"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 UEFA 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 the October 9, 2016 - 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."
Read the full article on eurosport.co.uk: O'Neill calls Koeman 'master tactician of the blame game', Koeman responds