Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
Muller and Kurz, along with his assistant Gunther Gorenzel, were relieved of their duties, with Markus Gisdol taking charge of first-team affairs with immediate effect.
Kurz has been in charge for just three months after succeeding Markus Babbel in January but failed to improve the team's fortunes.
Hoffenheim are second from bottom in the Bundesliga having won just two of Kurz's matches in charge.
The Baden-Wurttemberg club are four points from a relegation play-off spot and nine adrift of safety.
Kurz's final game in charge was a 3-0 defeat at Schalke on Saturday.
Gisdol, who will be presented this afternoon, will be on the bench for Friday's home game with Fortuna Dusseldorf.