"We are all shocked," said the national team's general manager, Oliver Bierhoff. "We are lost for words."
Joerg Neblung, a close friend and adviser of the the 32-year-old Enke, confirmed that it was suicide.
The incident happened at a crossing near Enke's home in Neustadt-Eilvese.
Hannover president Martin Kind said the goalkeeper had been "unstable" recently.
In 2006, Enke's daughter Lara died of a rare heart condition when she was two. Enke is survived by his wife, Teresa, and an eight-month-old daughter they had adopted.
Enke had been diagnosed with a bacterial stomach virus several weeks ago and did not play for Hannover for two months. Nonetheless, he was considered a strong candidate to start for Germany at the 2010 World Cup.