Klopp has been working as a pundit for German television at the World Cup.
He has repeatedly been linked with a return to club management, most notably with Real Madrid in March, which he said was “all nonsense”.
Klopp led Liverpool to seven major honours, including the 2019-20 Premier League title and the Champions League in 2019.
He previously led Borussia Dortmund to two German Bundesliga titles between 2008 and 2015, and was in charge of Mainz 05 for seven years, the club where he spent virtually all of his playing career.
Watzke, 69, worked closely with Klopp at Dortmund, where he was chief executive from 2005 until he stood down at his own request last year.
That relationship is thought to be one of the key factors in conducting such speedy negotiations with one of the most sought-after coaches in the game as Germany look to recover from a dismal World Cup display.
Germany were knocked out of this summer’s tournament in the round of 32, losing to Paraguay on penalties, continuing a run of below-par performances at major tournaments.
After Joachim Low led Germany to their fourth World Cup win in 2014, they failed to get beyond the group stage in 2018 and 2022, with Hansi Flick in charge for the latter campaign.
Nagelsmann succeeded Flick in September 2023, initially on a contract to the end of Euro 2024, which Germany hosted.
They reached the quarter-finals before losing in extra-time to eventual winners Spain.
Nagelsmann’s contract was later extended to this year’s World Cup and then to Euro 2028, but he stepped down after his second major tournament.