Reds chairman Peter Ridsdale has insisted Paul Hart's job is safe and criticised the behaviour of some fans after the defeat at Bradford City.
He said: "I understand that a number of our supporters are frustrated at the club's present league position,"
"This frustration is shared by many of us, but I do not see how anyone believes that the type of behaviour witnessed after the match at Bradford is likely to motivate individuals to get things right.
"Paul was upset by the incident and I firmly believe the use of such foul and abusive language is totally unacceptable,"
"We carry a huge responsibility to get things right for all of our supporters, but I still have absolute faith in Paul's ability to achieve our goals this season.
"We went into the campaign with high expectations and Saturday was a bad day at the office, but we had just gone four games unbeaten and have been playing some very good football.
"This is a very even league and fans have to give the new team time to gel. Paul was only hired in March and took on a major rebuilding job."
Regarding the behaviour of some Reds fans at Valley Parade, the Reds chairman said: "This complaint alleged that a number of Barnsley supporters used foul and abusive language throughout the match and, at the end of the game, used foul and abusive language towards the match officials, a number of our own players and our own manager,".
"We will continue to strive to get things right on the field. In the meantime, I would appeal to all Barnsley Football Club supporters to act in a responsible manner when following the team and when representing the club at away fixtures."