The letter’s writer – believed to be Brueckner – accuses prosecutors of perpetrating an “unbelievable scandal” by starting a “public prejudice campaign” against him.
The writer goes on to claim that prosecutors Hans Christian Wolters and Ute Lindemann have brought “shame” to the German legal system and demands their resignations.
The letter, dated 8 May, was reportedly accompanied by a hand-drawn cartoon of two people ordering “filet forensics” at a restaurant, perhaps in reference to Mr Wolters’ comment that there is no forensic evidence to suggest that Madeleine is dead.
Mr Wolters has also said that the child’s disappearance could be solved in months.
Madeleine, then three years old, disappeared in May 2007 from a Praia da Luz resort in the Algarve region of Portugal. Brueckner was living in a camper van nearby at the time.
He is alleged to have confessed to abducting Madeleine while talking to a criminal associate at a bar.
Brueckner denies being involved in Madeleine’s disappearance and has refused to talk to police.
Prosecutors say that they will interview him at the end of their investigation.
Mr Wolters said: “When we have completed our investigations, we will inform the public about the result and then also about further details.”
He also said that his team have received some “very interesting tips”.