Jorgensen's penalty miss saw Denmark's elimination from the tournament, following a 1-1 stalemate against Croatia.
The Danish FA released a statement and wrote on Twitter: "Stop. Our society must never accept death threats - neither against World Cup stars, politicians or others."
"It is completely unacceptable and indecent. We are reporting the case to the police to end this madness."
Three other Danish players missed spot kicks to seal their teams fate.
Jorgensen was the last to do so, and so received the majority of the abusive messages and threats on his Instagram and Facebook.
Jorgensen had been linked to a move to Newcastle United this summer but according to Sky Sports, the club would need to break their transfer record in order to acquire the Feyenoord striker.