The Times of London reported on Friday that Fischer, 53, would succeed current director Neil MacGregor at the museum, which draws around 6.7 million visitors every year.
Fischer, a Hamburg native and art historian, has been director of the Dresden City Art Collections since 2012 following a six-year stint at Essen's Folkwang Museum.
He has also worked in museums in Bonn and Basel, Switzerland, but never in the UK.
The last foreigner to head the British Museum was Italian-born Sir Anthony Panizzi, between 1856 and 1867.
Founded in 1753, the British Museum holds around eight million works from all over the world and throughout recorded history in its permanent collection, one of the largest in the world.
Fischer's appointment ends a five-month hunt for the right person to lead the museum after MacGregor announced he would be stepping down in April this year.
MacGregor is himself planning a move to Berlin to become an advisor to the German Culture Ministry on the Humboldt Forum, the rebuilt Prussian imperial palace at the heart of the German capital which is to become a museum housing treasures from Germany's monarchical past.