Mourinho is running out of time to save his job in North London (Tottenham).
Spurs have gone from the top of the league to ninth place in a matter of months, having won their last six Premier League fixtures.
Tottenham are currently nine points off Champions League places, while they also exited the FA Cup at the hands of Everton.
Mourinho now faces a decisive 12-day period in which Spurs play four games against Wolfsberger in the Europa League, then Burnley, Fulham and Crystal Palace in the Premier League.
A defeat in any of those matches could spell the end for Mourinho, with nine points the minimum expectation, although the former Chelsea and Manchester United coach does still retain the support of Daniel Levy for now.
Nagelsmann, 33, who dumped Spurs and Mourinho out of the Champions League 4-0 on aggregate last season, is regarded as one of the most exciting young coaches in Europe.
Chelsea approached the German before turning to Thomas Tuchel when they sacked Frank Lampard, but he did not want to leave Leipzig mid season.