QPR loanee Tom Carroll wants to "make it at Tottenham" but is prepared to consider all of his options at the end of the season.
Ilkay Gundogan has commited his future to Borussia Dortmund by signing a new contract.
The 43-year-old former Kaiserslautern boss will take up the post on January 1 after being named as the successor to Markus Babbel, who was sacked on December 3 after a poor run of form.
Kurz will be assisted by Gunther Gorenzel, with whom he also worked at Kaiserslautern.
Frank Kramer, who took caretaker charge of the club for two Bundesliga games after Babbel's departure, returns to his former role as boss of Hoffenheim's Under-23s.
"I said from the start that I was unconditionally at the disposal of the club," Kramer said on club website achtzehn99.de.
"I did that for the first team and I will continue to do the same for the Under-23 side.
"I will of course help the new manager with his tasks in any way that I can."
Kurz, a former defender, made over 300 Bundesliga appearances for Stuttgart, Nurnberg, Borussia Dortmund, Schalke and 1860 Munich, winning the league with Dortmund in 1995 and the UEFA Cup with Schalke two years later.
His managerial career began at Pfullendorf in 2005 before a spell at 1860 Munich.
He took charge of Kaiserslautern in 2009 and guided the club to promotion to the Bundesliga the following year. However, he was dismissed in March of this year with the club bottom of the league.
Kurz has a big job on his hands at Hoffenheim, who will spend the winter break seven points adrift of safety in the relegation play-off berth.
Babbel was dismissed after a 4-1 defeat to Werder Bremen but they have fared little better since.
Two defeats under Kramer have extended their losing sequence to six matches, and with just 12 points from 17 games, Hoffenheim are enduring their worst ever start to a Bundesliga campaign.