Interim manager Martin O'Neill will take charge for the final time in Wednesday's home game against Dundee before Nancy officially begins his term on Thursday.
Nancy joins Celtic from MLS club Columbus Crew, where he won the MLS Cup and Leagues Cup, and was named MLS Coach of the Year in 2024. He also won the Canadian Championship in his first managerial role at CF Montreal.
"I am so happy to be named Celtic Manager, it is a massive honour for myself and my family," Nancy said in a club statement.
"I know the history, I know the values of Celtic and I know what is expected of me on this journey."
Brendan Rodgers resigned at the end of October, his second spell at the club coming to an end after Celtic lost 3-1 to league leaders Hearts, which left them eight points off top spot, and they also went out of the Champions League to Kairat Almaty.
Former boss O'Neill came in as interim manager, and four consecutive league wins took Celtic to within two points of Hearts with a game in hand.
"I want to add my heartfelt thanks to Martin for all he has done in recent weeks," Celtic's principal shareholder Dermot Desmond said.
"He was the man we hoped could bring us through these last few weeks positively."
Nancy's first game in charge will be at home to Hearts on Sunday.