For the second straight season, the Bavarians crashed out of Europe's premier club competition at the quarterfinal stage after they could only manage a 1-1 draw at home to the Yellow Submarines, having lost the first leg 1-0 in Spain.
The result means Bayern's only realistic chance of lifting a major silverware this term is the Bundesliga, where they hold a nine-point advantage over Borussia Dortmund.
They were knocked out of the DFB-Pokal earlier this season in the second round after losing 5-0 to Borussia Monchengladbach.
Nagelsmann is in his first season at the Allianz Arena, having taken over from Hansi Flick at the start of the season, and is already discovering the pressure that comes with managing Germany's biggest club.
"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."
"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."
