The 18-year-old, who has yet to play for City's first team, will gain valuable experience for the Superliga side who currently lie in 6th position.
The Premier League champions have a link to the Danish club through youth team coach Scott Sellars, who played for the club in 2001.
Meanwhile, Oduro could be joined in the Superliga by team mate Mohammed Abu.
The 21-year-old midfielder is reportedly about to join champions Nordsjaelland on loan.
The midfielder has spent the first half of the season in La Liga with Rayo Vallecano but could now be on his way to Denmark to help Nordsjaelland close the gap on FC Copenhagen at the top of the table.