In this article, we study the Ricci soliton on quaternion bi-slant submanifolds of quaternion Kaehler manifolds. We obtain a lower-bound-type inequality in terms of expanding gradient Ricci solitons with a gradient-type vector field for the quaternion bi-slant submanifold of quaternion Kaehler manifolds. Additionally, we derive a series of lower-bound-type inequalities for semi-slant submanifolds, CR-submanifolds, hemi-slant submanifolds, slant submanifolds, invariant anti-invaraint submanifolds and totally real submanifolds in the same quaternion Kaehler manifolds. Finally, we discuss a double inequality for expanding gradient Ricci solitons on submanifolds in quaternion Kaehler manifolds and extend the same double inequality in terms of gradient Ricci solitons with a scalar concircular field on semi-slant, quaternion CR-submanifolds of quaternion Kaehler manifolds.