In this article we prove a Wiener Tauberian (W-T) theorem for L^p(G/K), p in [1,2), where G is one of the semisimple Lie groups of real rank one, SU(n, 1), SO(n, 1), Sp(n, 1) or the connected Lie group of real type F₄,and K is its maximal compact subgroup. W-T theorem for noncompact symmetric space has been proved so far for L¹(SL₂(R)/SO₂(R)) where the generator is necessarily K-finite ([S]). We generalize that result to the case of L^p functions of real rank one groups, without any K-finiteness restriction on the generator. We also obtain a reformulation of the W-T theorems using Hardy's theorem for semisimple Lie groups.