Barrios injury angers Dortmund
Barrios injured his thigh muscle during the Copa America final against Uruguay on July 24 when he came on as a substitute in the 76th minute. Paraguay lost 3-0.
"The tear in the muscle is much greater than we were told," Klopp told Borussia's website. "It's a very unhappy story."
"The forecast is that he will be out of action for five to six weeks. That is very, very annoying and will cast a shadow over our relationship with the Paraguayan federation."
Borussia begin the defence of their title at home to Hamburg SV on Friday when the Bundesliga kicks off.
The 26-year-old Barrios has scored 35 goals in 65 league appearances in his two seasons at Dortmund.