The Norwegian joined the club from Strømsgodset and won the title and League Cup in his first season in charge.
His reign had, however, been blighted by a series of poor performances in Europe and in domestic cup competitions.
Sunday’s Scottish Cup defeat to Rangers on penalties only increased the pressure on Deila.
He told the club’s website: “It is vital that the Club comes first and instead of me being the focus, hopefully now the team and the Club can be the focus as we enter this final important period of the season.
“It was an absolute privilege to be named manager of Celtic, such a wonderful football club, and I have enjoyed my time here immensely.”
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