Nagelsmann, whose sacking in March 2023 led to Tuchel's arrival, was tipped to return to Bayern, but has now extended his contract with Germany until 2026
Harry Kane scored a hat-trick in Bayern Munich's 8-1 demolition of Mainz 05 on Saturday to set a Bundesliga record as the champions cut the gap on leaders Bayer Leverkusen to seven points.
Kane, who joined this season for a league record 100 million euros from Tottenham Hotspur, became the first player to score at least twice in eight different games in his first Bundesliga season.
He also equalled Uwe Seeler's record for most goals in a debut Bundesliga campaign by lifting his tally to 30.
The England captain also netted twice in Tuesday's 3-0 win over Lazio which took Bayern into the Champions League quarter-finals.
Bayern are on 57 points in second place in the standings ahead of Leverkusen's home game against VfL Wolfsburg on Sunday.