The 23-year-old Borussia Moenchengladbach striker spat at Posch from close range when he confronted him off the ball during his side’s 2-1 defeat on Saturday in the German Bundesliga.
"Five of the six matches are to be served directly in Bundesliga and cup games, with the sixth suspended until 21 December 2021," said a statement from Germany’s football association.
Moenchengladbach have also docked Thuram a month's wages. The sum - around 150,000 euros - will go to charity.
"Marcus has accepted this and has also offered to engage with this social cause on his own behalf," said Monchengladbach's sporting director Max Eberl.
"I have come to know him as a reflective person with a great demeanour."
Pedigree
Thuram - the eldest son of France World Cup winning defender Lillian Thuram - made his international debut on 11 November in the friendly against Finland.
He featured three days later as a second-half substitute in the 1-0 win in Portugal which secured France’s berth in the semi-final of the Nations League. Thuram won his third cap in the 4-2 victory over Sweden on 17 November.
In an apology posted on Instagram, Thuram said: "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. Of course, I accept all of the consequences of my gesture.”
Moenchengladbach fell to eighth in the Bundesliga after the 2-1 home defeat.
