The departure of the 28-year-old suggest the managerless Reds are close to replacing Markus Schopp, and that the new man will be given the backroom support the Austrian lacked.
Eckert was appointed as head coach Gerhard Struber's assistant in August 2020 and also served under his successors, Valerien Ismael and Schopp.
Joseph Laumann was put in caretaker charge when Schopp was sacked after one win in 17 matches. But Eckert was unable to attend either match - one won, one lost - Laumann has taken charge of to date or the training sessions to prepare for them as he was quarantining with Covid-19.
Barnsley are expected to continue their recent tradition of appointing overseas coaches during the international break. Assuming they do, the new man's first game in charge will be at Fulham on Saturday.
Barnsley have not had an English head coach/manager since Paul Heckingbottom left in February 2018. Five men have done the job since.
Chief executive Khaled El-Ahmad has acknowledged Schopp's difficult task in succeeding Ismael was made harder by not being able to bring in his own coaches.
Eckert's departure frees up some space but a new goalkeeping coach, Thomas Fawdry, was only appointed last month.