The 22-year-old Abu joined City in 2010, but has been out on loan almost ever since, and made his professional debut with Stromsgodset during his first of two spells with the club.
He has also played for Eintracht Frankfurt, Rayo Vallecano, Lorient and Aarhus, and made his international debut for Ghana in 2011, but never appeared at senior level for City.
"MOHAMMED ABU: £MCFC can confirm that the Ghana international has agreed a permanent move to Norwegian Champions Stromsgodset. Good luck!" City said on Twitter.