Nicholson became the club's third chief executive of 2021 on September 10 after Dominic McKay departed suddenly following just 10 weeks in the role.
McKay had arrived from the Scottish Rugby Union as the long-serving Peter Lawwell retired, but his departure for "personal reasons" was announced before he returned to the oval-ball game the following month as chairman of European Professional Club Rugby.
Nicholson has been in an acting role ever since and has been planning for the January transfer window with manager Ange Postecoglou and recruitment staff.
The 45-year-old has been with Celtic since 2013, initially arriving as company secretary and head of legal, before being promoted to the role of director of legal and football affairs two years ago.
Nicholson told the club's official website: "As a Celtic supporter all my life, I will do everything I can to drive the club forward, working at all times in the best interests of Celtic.
"I have been fortunate to be part of this great institution for a number of years and it is a privilege to now hold this new responsibility.
"We have a team of brilliant people who are absolutely committed to Celtic and we will work together to deliver success for the club and our supporters.
"I look forward to working with Ange, the board, our colleagues and supporters in order to advance as a club, proud of our traditions and dedicated to constant improvement.
"The current environment remains uncertain and everyone at the club recognises and sincerely appreciates the ongoing commitment of our supporters.
"Our League Cup victory at the weekend gives us all something to celebrate and we look forward to continuing success for Celtic Football Club."
Celtic chairman Ian Bankier said Nicholson was "well known and highly respected across football domestically and internationally".
"He has the ability and experience to take Celtic forward positively and achieve great results," Bankier added.
Another current board member, Chris McKay, will assume the role of chief financial officer.
Bankier said: "The appointments of Michael and Chris bring important continuity.
"They have been close to Ange since his arrival as football manager in the summer, and since then they have supported him brilliantly and will continue to do so as we further develop our football operations."