The letter's authors, who describe themselves as "anti-German, anti-fascist flood brigades," threaten to attack five specific sections of the dam in order to "harm people across the country."
One of the dams being threatened is in Groß-Rosenburg, south of Magdeburg, where the River Saale flows into the Elbe, Der Spiegel reported. This dam broke on Sunday morning, though it was already under extreme pressure - it is currently impossible to tell whether the damage was down to sabotage.
The crisis team monitoring the anti-flooding measures is taking the threat extremely seriously, and air surveillance of the dams named in the letter has already been beefed up. Patrols have been told to report any irregularities directly to the police.
Investigators told Der Spiegel they had no information on who could be behind the letter, but that its content suggests left-wing extremists.
Thousands of emergency workers, troops and volunteers in Germany battled Sunday against central Europe's worst floods in over a decade, which have forced mass evacuations and which one politician termed a "national catastrophe."
The focus was on Magdeburg, where vast outlying areas were covered in a sea of brown water, sparked by recent torrential rains which have washed down the Elbe river system from the Czech Republic.
The water level in Magdeburg reached 7.45 metres (24 feet) in the morning, vastly higher than the usual level of around two metres and worse than massive floods that struck the region in 2002, local authorities said.
Almost 3,000 residents were evacuated from the city's Rothensee district, where hundreds of army troops were working to reinforce a dyke protecting a crucial electricity facility to prevent a wider power outage in the city.
More townships were evacuated around Barby, where the Saale tributary meets the Elbe. Some of the 8,000 residents of the nearby town of Aken were taken to safety on military armoured personnel carriers and ambulances.
President Joachim Gauck was due to visit the flood-hit states of Saxony and Saxony-Anhalt on Sunday and join a church service with emergency crew and residents in the town of Halle.