According to BILD, the Interior Ministry in Dresden confirmed that police have identified six protesters who gave the salute.
In two cases, the Attorney General's Office has applied for accelerated procedures at the Chemnitz district court.
“It is now important that the accelerated procedures be applied swiftly to quickly convict and punish the perpetrators. The rule of law must show that it is capable of acting,” said Saxony's Interior Minister Roland Wöller (CDU).
Investigators had been probing cases against far-right protesters who gave the illegal Hitler salute during unrest in Chemnitz, following the fatal stabbing of a man allegedly by two immigrants. Police are searching for a third suspect.
Footage and photographs from Chemnitz showed far-right demonstrators raising their hands to make the Nazi salute.
The swastika and other Nazi symbols are banned in Germany. Those caught breaking the law can be fined or face a jail term of up to three years.
The first cases of the six suspects could be scheduled as early as next week. The charges must first be served, said the spokeswoman for the district court Birgit Feuring.
“The law requires that an accelerated process be completed within six weeks,” said Feuring.
The first cases involve two suspects: a 32-year-old and a 34-year-old from Chemnitz. They are said to have publicly shown the salute at a rally on August 27th in Chemnitz, the General Prosecutor's Office reported.
One of the men is also accused of insulting a police officer several times.
The incident in Chemnitz follows the suspension of two police officers in Germany’s southern state of Bavaria over allegations that they performed the illegal Nazi salute.
In a statement released over the weekend, police said the two officers and a third man, believed to be a security guard, were accused of shouting anti-immigrant remarks and performing the salute at a pub in the city of Rosenheim last week.
The allegations were made by a witness who was in the same beer garden as the officers, the statement said.
After the footage in Chemnitz emerged, Foreign Minister Heiko Maas said: “If once again people are parading today in the streets making Nazi salutes, our past history forces us to resolutely defend democracy.”
He urged people to “get up off the sofa and speak up,” adding: “If the Hitler salute is made on our streets today once again, it is a disgrace to our country.”