The defender did not disclose the reason for the unsuccessful application and will now train with the Under-21 side as he waits to make his debut for his new team
The Croatia defender completed his protracted move from Southampton to Anfield on Sunday, signing a long-term deal with the club for an undisclosed fee.
With Brendan Rodgers and the first-team squad in the USA for the International Champions Cup, Lovren was expected to fly out and possibly make his first appearance in Wednesday's match with Manchester City.
However, the 25-year-old has revealed he will not feature in America, meaning he will likely have to wait until a week before the season starts to make his debut against Borussia Dortmund.
"Unfortunately, I could not immediately get a visa," he told Croatian newspaper Sportske Novosti. "Therefore I cannot join my new team-mates in the United States. I'm very sorry about that.
"Of course it's not good that I have not joined the team's preparations, however, I have plenty of days until their return from the tour.
"I will train with the Under-21 team in Liverpool, Brendan Rodgers will send me a programme for me to practice."