The source told Reuters that Balakov, 45, had agreed to join the club.
Kaiserslautern said there would be a news conference to unveil a new coach on Thursday, who would also take over the afternoon training session, but refused to reveal the name of Kurz's successor.
Balakov, who played for VfB Stuttgart from 1995 to 2003 and was briefly their player-manager, has been coaching Hajduk Split in Croatia since 2011.
He was a member of the Bulgaria team that finished fourth at the 1994 World Cup.
Kaiserslautern are bottom of the Bundesliga after failing to win in their last 16 league games and lie five points below the relegation play-off spot.
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