The Switzerland international, who came through the ranks at FC Zurich, moves to the City Ground for the remainder of the 2018-19 campaign.
Janko joined Manchester United in 2013 and was voted Reserve Team Player of the Year in his first season but went on to only make one senior appearance for the club.
The 22-year-old enjoyed a loan spell at Bolton Wanderers in 2015 before completing a permanent move to Celtic later that summer.
Janko, a right-back who can also play on the right wing, only joined the Portuguese side earlier this summer from Parkhead.
Forest sit 13th in the Championship table with one win and four draws.