A large class of noncommutative spherical manifolds was obtained recently from cohomology considerations. A one-parameter family of twisted three-spheres was discovered by Connes and Landi, and later generalized to a three-parameter family by Connes and Dubois-Violette. The spheres of Connes and Landi were shown to be homogeneous spaces for certain compact quantum groups. Here we investigate whether this property can be extended to the noncommutative three-spheres of Connes and Dubois-Violette. Upon restricting to quantum groups which are continuous deformations of Spin(4) and SO(4) with standard coactions, our results suggest that this is not the case. (c) 2005 American Institute of Physics.