Global optimization of the geometry of ${\mathrm{Al}}_{13}$ cluster using the density-functional theory and generalized gradient approximation yields two nearly degenerate isomers having Jahn-Teller distorted icosahedral and decahedral structures. As these two isomers of ${\mathrm{Al}}_{13}$ interact with alkali-atoms X $(X=\mathrm{Li},$ Na, K, Rb, and Cs), the Jahn-Teller distortions in the bare cluster isomers disappear in all cases except in Cs. The binding energy of alkali atoms, X to ${\mathrm{Al}}_{13}$ systematically decreases from Li to Cs for both the isomers. This is shown to result from a competition between the size and the ionization potential of the alkali atoms. In addition, the difference in the total energies between icosahedral and decahedral structures containing the alkali atoms becomes smaller than that between the bare ${\mathrm{Al}}_{13}$ isomers. The vertical ionization potentials of ${\mathrm{Al}}_{3}X$ is larger than that of ${\mathrm{Al}}_{13}$ in the decahedral structure while the opposite is the case with the icosahedral isomer. The above results based on a frozen core approximation were repeated using an all-electron basis. While there are quantitative differences between these results, these are negligible at the present level of theory.