In this article, we consider the two-dimensional stochastic Navier-Stokes equation (SNSE) on a smooth bounded domain, driven by affine-linear multiplicative white noise and with random initial conditions and Dirichlet boundary conditions. The random initial condition is allowed to anticipate the forcing noise. Our main objective is to prove the existence and uniqueness of the solution to the SNSE under sufficient Malliavin regularity of the initial condition. To this end we employ anticipating calculus ideas.