Dann will miss the Sky Blues' remaining three Championship fixtures. The centre-half was forced off with a groin injury midway through the second half of the 1-0 defeat at Nottingham Forest on Saturday.
Morrison, who moved to the Ricoh Arena from Crystal Palace last summer, is Coventry's leading scorer this season with 11 goals and captained the side earlier this season when Dann was again sidelined through injury.
"I have to do a job. As I am in the starting eleven I will never complain and I will always be happy to do well and give my best for Coventry," Morrison told BBC Coventry & Warwickshire.