Work fluctuation and total entropy production play crucial roles in small thermodynamic systems subject to large thermal fluctuations. We investigate a trade-off relation between them in a nonequilibrium situation in which a system starts from an arbitrary nonequilibrium state. We apply a variational method to study this problem and find a stationary solution against variations over protocols that describe the time dependence of the Hamiltonian of the system. Using the stationary solution, we find the minimum of the total entropy production for a given amount of work fluctuation. An explicit protocol that achieves this is constructed from an adiabatic process followed by a quasistatic process. The obtained results suggest how one can control the nonequilibrium dynamics of the system while suppressing its work fluctuation and total entropy production.