Off-axis lensless holography is considered with a sinusoidal phase modulation at the object plane. The variational algorithm for phase and amplitude reconstruction is based on the algorithm proposed in the paper 1[V. Katkovnik, I. A. Shevkunov, N. V. Petrov, and K. Egiazarian, “Wavefront reconstruction in digital off-axis holography via sparse coding of amplitude and absolute phase”, Opt. Lett. 40, 2417-2420 (2015)]. The forward wavefront propagation is modelled using the Fourier transform with the angular spectrum transfer function. The multiple intensities (holograms) recorded by the sensor vary in dependence to the angle of the phase diffraction grating. The i.i.d. Gaussian noise is added to observations to make them closer to real experimental conditions. The root mean square error (RMSE) values of the phase reconstructions were compared in two scenarios: with and without the diffraction grating. Computational experiments showed that with sinusoidal phase modulation RMSE values are decreased about 20%. These results support the conclusion on advantage of the proposed phase modulation gratings in off-axis lensless digital holography.