The development of rolling texture was studied by an X-ray diffraction method using pure aluminum single crystals with an initial crystal orientation deviated by 5 degrees about the rolling direction from the ideal (001) [100] orientation. The crystal was cold-rolled with constraint provided by a polycrystalline aluminum frame. The rolling texture was compared with the texture measured for a different crystal rolled without constraint. The texture rolled without constraint was also compared with the texture of (001) [100] single crystals rolled without constraint reported in our previous study. At 50% reduction in thickness, the texture of the crystal deviated by 5 degrees from the ideal orientation rolled without the constraint was composed of two main orientations, which were characterized by resultant rotations about the transverse, rolling and normal direction axes with respect to the initial orientation. The two main orientations reached {123} [634] at 95% reduction in thickness without passing through {101} [001] and {110} [112]. In order to realize crystal rotation from {100} [001] to {123} [634] in rolling, the {100} [001] single crystal should have the initial orientation deviated by 5 degrees from the ideal orientation, and in addition be rolled without constraint. At 95% reduction in thickness, different character of textures was observed in the crystals deviated by 5 degrees from the ideal orientation rolled with and without constraint. In both cases, however, no cube orientation was recognized in the textures.