A class of partially coherent elliptical sources with twisted Laguerre Gaussian Schell-model (TLGSM) correlation function is proposed, which are capable of producing beams whose intensity profiles may vary substantially. This kind of beam can be viewed as the generalization of the LGSM beams. Properties of the spectral density during propagation in free space and atmospheric turbulence are investigated with varying quantities related to the beam source and the medium. It is shown that the elliptical TLGSM beams evolve in a manner that is much more complex compared to the LGSM beams. In addition, the behaviour of the rotation angle is further analysed by quantitative examples.