Using symmetry properties, we determine the mixing pattern of a class of nonlocal quark bilinear operators containing a straight Wilson line along a spatial direction. We confirm the previous study that mixing among the lowest dimensional operators, which have a mass dimension equal to three, can occur if chiral symmetry is broken in the lattice action. For higher dimensional operators, we find that the dimension-three operators will always mix with dimension-four operators, even if chiral symmetry is preserved. Also, the number of dimension-four operators involved in the mixing is large, and hence it is impractical to remove the mixing by the improvement procedure. Our result is important for determining the Bjorken-x dependence of the parton distribution functions using the quasi-distribution method on a Euclidean lattice. The requirement of using large hadron momentum in this approach makes the control of errors from dimension-four operators even more important.