The 32-year-old Hannover player was confirmed dead at the scene by German police, with early indications suggesting probable suicide.
"We are all shocked," said Germany's general manager Oliver Bierhoff. "We are lost for words."
The president of the German Football Association, Dr Theo Zwanziger, said the nation's football community was "deeply distressed and in mourning".
Enke lost his two-year-old daughter Lara to a rare heart condition in 2006. He leaves behind his wife, Teresa, and their adopted eight-month-old daughter.
- Football.co.uk staff
Source: DSG