Would you please your question more clearly

No, it is not : computing the displacement gradient from the strain tensor requires the intermediate computation of the anti-symmetric part of this gradient (noted w).

It requires (i) derivating once more the strain tensor, (ii) integrating some combinations of its partial derivatives to obtain w (iii) summing : \grad u = \eps + w (you should find the details in some textbooks of solid mechanics / elasticity)

At my knowledge, you cannot perform steps (i) and (ii) with the sole knowledge of the strain tensor at some Gauss points.

By curiosity : why do you need to access this gradient ?

Barhli did it: https://www.academia.edu/27199939/Obtaining_the_J-integral_by_diffraction-based_crack-field_strain_mapping