The 23-year-old complained about his lack of playing time for the German champions this season and also accused his coach of favouring certain players.
"I have realised Ivan's not happy with his situation and it would be strange if he was," said Klopp. "That's perfectly normal.
"And for the future, not only for Ivan, whoever believes he could up the pressure with making those statements is on the wrong track, that's nonsense.
"I have so far not been able to draw the great potential he has out of him and he has not been able to do so as well. It's my responsibility to help the boys to get their potential onto the pitch."
Perisic joined Dortmund from Club Brugge in 2011.