The 25-year-old was injured in the first half of Derby's 2-0 win over Stoke City in the Championship (second tier) on Monday and was taken off on a stretcher.
"The club will provide Patrick with the highest level of medical care and rehabilitation throughout his recovery," the club said in a statement.
"As a result of this injury, Patrick will unfortunately miss this summer's FIFA World Cup. At this stage it would be wrong to put a timeline on his recovery."
Agyemang has scored six goals in 14 internationals since making his U.S. debut last year. He played in both the team's friendlies during the March international window.
US coach Mauricio Pochettino must name his World Cup squad by May 30. The U.S., co-hosting the tournament with Canada and Mexico, will face Paraguay, Australia and Turkey in Group D.
The World Cup starts on June 11.