We consider solutions of stochastic differential equations which diverge to infinity as the time parameter goes to infinity. If the coefficients converge as the spacial variable goes to infinity, then the solutions will get close to some Gaussian processes with positive drifts as the time parameter goes to infinity. In this paper, we prove Berry-Esseen type bounds for the solutions in this setting. In particular, we obtain bounds of the total variation distance between the law of the centered and scaled solutions of the stochastic differential equations and the standard normal distribution with an optimal rate of convergence in the time parameter. In the proof we apply the Malliavin-Stein method to estimate the total variation distance.