Thuram saw red in the 78th-minute after he spat at Hoffenheim's Stefan Posch and is likely to be hit with a length ban by the German Football Association for his actions, where consequences are heightened due to the ongoing coronavirus pandemic.
Following the match, Gladbach manager Marco Rose condemned the incident and said it was "beyond anything," and Gladbach have confirmed a month's fine has been issued to the France international.
"I had a long discussion with Marcus this morning, in which he apologised to me and to the club once again for his actions," Gladbach sporting director Max Eberl said. "We have known Marcus for almost two years, we know his background and his family.
"Yesterday's actions do not represent that. Marcus is devastated and has assured me that he did not purposely spit at Stefan Posch. He told me that during a dispute with Stefan Posch, he had unintentionally spat after cursing several times in French in the heat of the moment.
"He knows that this does not change anything and that the pictures speak for themselves, and that the sending off was the right call. I believe Marcus because I have come to know him as a reflective person with a great demeanour.
"He did it, but he did not do it with intent. As a club, we will fine him a sum equivalent to one month's wage for his behaviour that led to the dismissal, which will be donated to a good cause.
"Marcus has accepted this and has also offered to engage with this social cause on his own behalf. Last night, Marcus apologised to Stefan Posch, TSG Hoffenheim, his teammates, the coaching staff and the Gladbach fans, and I would like to do this again explicitly on behalf of Borussia Monchengladbach.
"Marcus made a big mistake and is being punished for it. He remains the same person we know, and we will stand by him."
Thuram, son of France's 1998 World Cup winner Lilian Thuram, also apologised for his action on Twitter. He wrote: "Today something took place that is not in my character and must never happen.
"I reacted to an opponent in a wrong way and something occurred accidentally and not intentionally. I apologise to everyone, to Stefan Posch, to my opponents, to my team mates, to my family and to all those who saw my reaction.
"Of course, I accept all the consequences of my gesture."