Let X, $X_1$, $X_2$, ... be i.i.d. random variables with zero means, variance one, and set $S_n;=;{sum}^n_{i=1};X_i$, $n;{geq};1$. Gut and $Sp{check{a}}taru$ [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If P($|S_n|;{geq};{varepsilon}{sqrt{2n;{log};{log};n}}$) is replaced by E{$|S_n|/{sqrt{n}}-{varepsilon}{sqrt{2;{log};{log};n}$}+ in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when X belongs to the domain of attraction of the normal law.