(Updates with O'Neill statement)
Everton, thumped 4-0 by Liverpool at Anfield last season and beaten 1-0 by their neighbours at Goodison Park in December, have failed to win at their bitter rivals since 1999.
"I don't know why. I heard in the last two seasons they were too afraid to play against Liverpool. Why you need to be afraid, I don't understand that," Koeman told reporters on Friday.
Koeman was also critical of Ireland manager Martin O'Neill's handling of James McCarthy after the midfielder aggravated a hamstring injury while on international duty this month.
"The assessment of the Ireland medical team would be that it would be high risk for James to play. When asked to play he said he felt he was fit," Koeman added.
"In my opinion the Ireland manager in this instance was not protecting the player. I spoke to James, he needs to take responsibility... It is not the first time."
O'Neill responded by calling Koeman the "master tactician of the blame game", saying that a review of Everton's pre-season programme might "provide some enlightenment".
In a statement, the Ireland manager said McCarthy played his last game at Euro 2016 in late June before returning to Everton after a short break.
"James last played for his country on 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."
Koeman hit back on twitter, saying: "James McCarthy began his pre-season three and a half weeks after Ireland were knocked out of the Euros. From the master tactician."
Everton will also be without France midfielder Morgan Schneiderlin, Argentina defender Ramiro Funes Mori and Ireland right back Seamus Coleman for the Anfield game because of injuries suffered on international duty. (Reporting by Shravanth Vijayakumar in Bengaluru, editing by Ed Osmond)