This paper is devoted to the classification of all flag-transitive point-primitive non-trivial $2$-$(v, k, λ)$ designs with the alternating group $A_n$($n \le 10$) as the socle of their automorphism groups, and 87 different designs are obtained up to isomorphism. The results of this study further improve the classification theory of designs under the action of almost simple groups, and provide reference for the follow-up study of similar problems.