The Scottish champions confirmed the Frenchman’s departure in a brief statement on Monday, announcing that his contract had been terminated with immediate effect.
Nancy’s backroom staff -- Kwame Ampadu, Jules Gueguen and Maxime Chalier -- have also left the club, while Paul Tisdale has left his role as head of football operations.
Nancy leaves Celtic second in the Scottish Premiership, six points behind leaders Hearts and level with Rangers, but his position had become untenable after a run of six defeats in eight matches.
The former Columbus Crew boss managed just two victories, with defeats including a League Cup final loss to St Mirren and setbacks against both Hearts and Rangers in the league.
His dismissal comes just two days after Celtic were beaten 3-1 by Rangers at Parkhead, a result that saw pressure on the 48-year-old intensify.
Celtic said a further update would be provided "as soon as is practical", with the club expected to move quickly to appoint an interim successor. The names of former managers Martin O’Neill and Ange Postecoglou have been raised in early discussions.
Celtic are next in action against Dundee Utd on 10 January.