The Arsenal forward lasted only 30 minutes before being replaced by Younes Ebnoutalib after appearing to suffer a hamstring injury.
Havertz went down following a heavy challenge from Dutch midfielder Quinten Timber, the brother of his Arsenal team-mate Jurrien Timber. He initially tried to continue but was eventually forced off after appearing to feel discomfort at the back of his thigh.
Klopp confirmed after the match that Havertz and Jamal Musiala, who picked up a thigh problem during the warm-up, would both undergo MRI scans.
"Both will undergo MRI scans tonight and then we'll have the diagnoses," Klopp said.
"They're probably out for us, but I hope they'll be back soon (for their clubs)."
The timing of the injury will concern Arsenal, with Havertz expected to return to club duty after the international break.
Arsenal host Leeds in the Premier League on October 10, with the club now waiting for the results of Havertz's scan to discover whether he will be available.
Erling Haaland made history with a two-goal display as Norway beat Denmark 3-2 in the Nations League, taking him past Zlatan Ibrahimovic as the highest-scoring Nordic men's international player.
Netherlands manager Xavi has recalled Manchester United striker Joshua Zirkzee after Brian Brobbey was ruled out of the remainder of the international break with a muscle injury.
Jurgen Klopp was left encouraged by Germany's performance in his first match as head coach, despite seeing his side concede a late equaliser in a 1-1 draw with the Netherlands in the UEFA Nations League on Thursday.
Liverpool defender Virgil van Dijk believes the Netherlands can make significant progress under new head coach Xavi Hernandez after beginning his tenure with a 1-1 draw against Germany.
