A usual way of approximating Hamilton-Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size
for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on the Mickens' rule of nonlocal approximation. The scheme obtained in this way is
unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided.