Bundesliga leaders Bayern, who stand on the brink of a ninth successive title, are set to lose head coach Hansi Flick at the end of the season after he asked to be released from his contract.
While Flick is seen as a possible successor to Joachim Low with the Germany national team, the 33-year-old Nagelsmann is considered one of Europe's best young coaches.
His name has been linked with the likes of Real Madrid, Manchester United, Tottenham and many others at various times, but Bayern have also kept a close eye.
Bayern are said to be chasing a fee for the release of Flick to the German Football Federation (DFB), while it has been reported that Leipzig want up to €30million ($36m) to free up Nagelsmann.
Bayern Munich priority as next manager is Julian Nagelsmann, he wants to join Bayern as first choice - negotiation still not easy with RB Leipzig, Bayern are intentioned to push in the next days.— Fabrizio Romano (@FabrizioRomano) April 26, 2021
No talks with Massimiliano Allegri. 🔴 #FCBayern https://t.co/JzTTLOCvTl
Bavarian newspaper TZ puts the fee being sought in the region of €20million, while Sport1 suggests the higher sum was quoted by Leipzig chief executive Oliver Mintzlaff after an initial approach from Bayern.
Nagelsmann's contract with Leipzig, who sit second in the Bundesliga, runs until the end of the 2022-23 season.
The former Hoffenheim coach joined Leipzig in 2019 and helped the team reach the semi-finals of last season's Champions League, the competition that Bayern won to complete a famous treble under Flick.
Nagelsmann has yet to declare his wishes amid the discussions about his future.
Former Bayern and Germany star Lothar Matthaus recently suggested Nagelsmann was a shoo-in for the Allianz Arena job.
Nagelsmann has won 53, drawn 21 and lost 16 of his 90 matches in charge of Leipzig.
Speaking on April 18, he said there had been no talks with Bayern.
"There is simply nothing I can report," Nagelsmann told a media conference on that day. "I think it's totally wrong to then sit down and say what if, and maybe get yourself talked about somewhere and make yourself important on things that aren't there.
"I can't sit down and say, I'm going to break up with Lena Gercke [German model] tomorrow, because I've never dated her, for example.
"I'm sorry to say that there are no new developments compared to the last press conference. The same things apply that I said there.
"There are and were no discussions and no offer. That's why I don't know what Bayern's wish is. I have said I don't want to start a war with my employers - that's still the case."