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  Koeman has advised midfielder James McCarthy not to join the Ireland squad after previously saying he had been "massively overloaded" by the Republic.
  Republic manager Martin O'Neill said this week that Koeman was "bleating" about the matter.
  Keane added: "Maybe Everton need to look at their own training schedule."
  McCarthy, 25, has been ruled out of the Republic's World Cup qualifier against Austria on Saturday because of a hamstring injury.
  He has missed most of his club's season because of injury but played in the Republic's qualifier wins over Georgia and Moldova last month.
  That angered Everton boss Koeman, who has only seen McCarthy play 33 minutes of competitive action for the Premier League club since then.
  However, Keane says he "could not care less" about Koeman's concerns - and also criticised previous Toffees manager Roberto Martinez.
  "Everton traditionally have always had brilliant Irish players doing well for the football club, so they shouldn't be so quick to stop Irish players coming to play for Ireland," said the former Manchester United midfielder.
  "We have had a problem previously with Martinez - I think he was slightly over the top.
  "Every time, we felt, the Everton players were turning up - and this is not a criticism of the players, this is more from Everton and their staff - they were always carrying knocks.
  "I always felt the Everton players were going to turn up on crutches or crawling in the hotel door, and now it looks like we are probably going to have that issue again with Koeman. I hope not.
  "Maybe Everton as a club need to - and it's a brilliant football club - maybe their players need to toughen up a little bit."
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target: Republic of Ireland assistant manager Roy Keane says Everton players must "toughen up" in an ongoing row with Goodison boss Ronald Koeman.