We calculate the quantum entanglement of ground-state electron in two spatial dimensions of rectangular GaAs/AlxGa1-xAs quantum wires (QWRs) with varying the wire width and the confinement potential. The energy, von Neumman and linear entropies of ground-state electron show rapid convergences as a function of the standing-wave basis-set size. The non-zero entropies indicate the existence of quantum entanglement in two spatial dimensions. With increasing the wire width, the von Neumann and linear entropies increase rapidly until they reach a maximum after which, they monotonically decrease. The linear entropy is much smaller than the von Neumann entropy, however they show a totally similar entanglement behavior. The maximum entropies increase gradually with increasing the confining potential. When either of QWR dimensions becomes sufficiently large, the entropies are almost zero. At this case, the electron state can then be separated into a product of two-dimensional components of the electron wave function. In addition, the probability density distributions have been investigated for a given set of values of the wire width and the confinement potential. The overlap of the barrier region lead to the electron entanglement in two spatial dimensions, which is proportional to the probability finding the electron in the corner of the barrier region.