Klopp, working as a lead pundit for German broadcaster Magenta TV, had already drawn criticism before Germany's opening match against Curacao for suggesting Nagelsmann should leave out Bayern Munich forward Jamal Musiala in favour of Deniz Undav.
It was a separate comment, made during an on-air exchange with fellow pundit Thomas Muller, that proved more contentious. Discussing Nagelsmann's position as head coach, the former Liverpool manager remarked that the Germany boss was "still" picking the team — phrasing widely read as a suggestion that his tenure might not last much longer.
Klopp has been one of the names most frequently linked with succeeding Nagelsmann at the helm of the national team, which gave the comment added weight.
Nagelsmann was asked about the remark before kick-off against Curacao and gave a pointed response, telling a reporter he was surprised the question had come from them before moving on.
Germany then produced a 7-1 win over Curacao, and it was in the aftermath of that result that Klopp chose to address the comment directly. Speaking on air, he described the word as the most regrettable of his year so far.
"I've already found the most hated word of the year: 'Still'."
He went on to explain how the remark had slipped out without him registering its implications at the time.
"I could have punched myself in the face for that, but it was already too late and I was on TV. It just slipped out so casually, and has absolutely no relevance."
Klopp added a self-deprecating note about his own track record with off-the-cuff remarks.
"What I've realised is, I'll be 59 on Tuesday and I'm still an idiot."
He then turned to Nagelsmann directly to offer reassurance about his intentions going forward.
"We are completely on your side, whatever you do with this. Nothing will come of it that is intended to disrupt the process here."
The episode adds to a difficult build-up for Nagelsmann off the pitch, even as his side delivered one of the most emphatic results of the World Cup's opening weekend on it. Klopp's standing as a prospective successor means every comment he makes about the national team will continue to be read through that lens, whether he intends it or not.
For Nagelsmann, a seven-goal win is the kind of response that quietens most noise. Whether Klopp's apology draws a line under the episode before Germany's next group fixture against Ivory Coast remains to be seen.