Germany lost 4-3 on penalties to Paraguay on Monday after a dour 1-1 draw in the last 32.
"I'm available if the DFB (German Football Federation) wants it, I have a deal until 2028. If not, then they have to tell me," Nagelsmann told broadcaster ZDF in a fractious interview.
"I am not the sort of person who runs away."
His contract runs until after the European Championship in the British Isles in two years. This was his first World Cup as coach after a creditable quarter-final exit at the home Euros in 2024 to eventual winners Spain.
Germany national team director Rudi Völler showed his support for Nagelsmann after the game.
"I'm still convinced that he's the right man for the job, the right person in the right place," he said.
"When you're eliminated like this, many people won't understand it. I still think he's an absolutely top-class coach," he added.
Germany's previous group-stage eliminations came in 2018 under 2014 World Cup-winning coach Joachim Loew and 2022 under Hansi Flick. Both were kept on after the shock exits.
This defeat is less of a shock given Germany's pedigree has fallen so far since 2014 and they at least got out of the group as winners.
The big talking point was the referee's decision to rule out Jonathan Tah's header in extra-time for minimal contact between Waldemar Anton and Paraguay keeper Orlando Gill.
Gill went down during the corner but was on the way back up again as the powerful header flew in. Moroccan referee Jalal Jayed was only alerted to the possible issue by the video assistant and made his decision after studying the monitor.
What had looked like a 2-1 lead and probable last-16 berth was suddenly snatched away, much to Nagelsmann's anger.
"It is a scandal that the goal can be ruled out. I have no idea what he saw. It is a joke," he rumbled.
"It is a total scandal, there is no way it is a foul."
However, Germany great Lothar Matthäus believes Nagelsmann era will come to an end.
"I believe we need to move forward with a new coach after this World Cup. It was just too much. There's probably a lot to work through, both on and off the field," he told the Bild newspaper.
"One game can always happen. But we already played terribly against Ecuador," he said.
13°C