Nagelsmann, 38, has been under fire since Germany's last-32 exit from the World Cup after a shock loss to Paraguay.
Showing pictures of Nagelsmann at German FA (DFB) headquarters in Frankfurt, Bild reported Thursday the coach had been offered a severance package of seven million euros ($8 million) to leave the position in a three-hour meeting.
Last renewed in January 2025, Nagelsmann's contract is set to run until 2028.
Figures on his annual salary have not been made public, although German media reports he earns roughly seven million euros per year.
Germany's loss on penalties to Paraguay marked the third straight early World Cup exit for the four-time champions, after group stage eliminations in Russia and Qatar.
Monday's defeat was Germany's first knockout match since winning the World Cup in Brazil in 2014.
On Tuesday, DFB president Bernd Neuendorf announced an immediate investigation into the World Cup failure.
A decision on Nagelsmann is expected "by the beginning of next week at the latest", AFP sports subsidiary SID reported Thursday.
Several German media outlets, including Sky and Munich daily Sueddeutsche Zeitung, have said Klopp is the overwhelming favourite to succeed Nagelsmann.
Klopp stepped away from management after leaving Liverpool in 2024 and has since been engaged as the head of global football for energy drink brand Red Bull, overseeing a multi-club network which includes clubs in Austria, Germany, Brazil, the United States and Japan.
FIFA World Cup 2026
The 2026 World Cup is taking place from June 11th to July 19th in the United States, Canada and Mexico. The tournament features 48 national teams and is played at 16 modern stadiums.
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