0401 GMT February 21, 2019
Hartwig Fischer, whose appointment has yet to be officially confirmed, would be the first non-British head of the institution since the 1860s.
Fischer is currently director general of the Dresden State Art Collections and was formerly director of the Folkwang Museum in Essen, BBC wrote.
Neil MacGregor steps down as director of the British Museum in December.
The museum is the most popular visitor attraction in Britain, welcoming 6.7 million people in 2014.
According to The Times, Fischer's appointment has been approved by the museum's trustees and will shortly be signed off on by Prime Minister David Cameron.
The last non-British head of the museum was the Italian-born Sir Anthony Panizzi, who held the post of principal librarian between 1856 and 1866.
Fischer began his museum career at the Kunstmuseum in Basel, where he was curator of 19th century and modern art from 2001 to 2006.
He has never worked in Britain but did co-curate a 2006 exhibition of Wassily Kandinsky's work that was shown at Tate Modern in London.
Fischer studied History of Art, History and Classical Archaeology in Bonn, Berlin, Rome and Paris and holds a doctorate in philosophy from the University of Bonn.
Born in Hamburg in 1962, the 52-year-old speaks German, English, French and Italian and is married to Ilaria Piqueras Fischer, a psychoanalyst.
Fischer was not among those initially linked to the post but was reportedly put forward as a candidate by MacGregor himself.