Bayern was eliminated from the UEFA Champions League in midweek after a 1-1 draw at home to Villarreal resulted in a 2-1 aggregate quarter-final defeat.
It means Die Roten can only win one major trophy in Nagelsmann's first season as head coach, though a nine-point lead at the top of the Bundesliga with five games to go means that trophy looks reasonably secure.
Nagelsmann said receiving threats is not out of the ordinary as he opened up on the abuse when previewing Bayern's weekend fixture with Arminia Bielefeld.
"I get them after every game, regardless of whether we win or lose. I only ever see the first line and then delete them all at once," he said.
"They even shoot at my own mother, who doesn't play football at all. That's a little wild.
"There are more death threats when we play a back three. How do I deal with it? I don't give a f***. I cannot understand. As soon as you turn off the TV, people forget their decency. But that's all useless. They think they're right, that's the bizarre thing.
"I don't think the club is increasing security. You also move as a private person. I don't want to provoke anyone now."
Club legend and former chief executive Karl-Heinz Rummenigge spoke about how uncertainty over the contracts of several big-name players may have proved a distracting factor in Bayern's European demise.
Nagelsmann says it is easier to accept criticism from such quarters.
"I am aware that you have to put up with criticism from all sides. That's normal, part of it. Karl-Heinz Rummenigge's criticism is manageable for me. I can handle that," he added.
"Maybe not quite as good with the 450 death threats on Instagram. But I don't read them all, of course that's a bit irrelevant.
"Of course, if you are eliminated in two out of three competitions, a coach will also be criticised. But I can take it and keep working."
Nagelsmann also stated he had held constructive talks with Bayern's hierarchy after the Villarreal setback.
"We sat together for two hours on Wednesday and talked about the game. I picked out the most important things again, but again our game was good. We lost it in the first leg," he said.
"I had a long phone call with [chief executive] Oliver Kahn yesterday, also about the squad and my ideas. He wants to have a picture of what the coach is thinking. The exchange has been very good so far.
"We are very good at planning, but implementation is not that easy. The squad planning changes every day, you imagine something. Two days later it looks very different. That's where the fast pace of business comes into play."