In this paper, we systematically study the dopability and magnetic properties of a 3d TM-doped atomic-thick SnTe(001) monolayer based on first-principles calculations. It is found that, for separately distributed TMs in a SnTe(001) monolayer, all of the TMs, except Sc, Cu, and Zn for the substitutional configuration and Ni, Cu, and Zn for the adsorption and interstitial configuration, could induce local magnetic moments. On the other hand, contradictive to the intuition that TM may adsorb on the SnTe(001) slab, substitution is more favorable. Even though the formation energy of Ni adsorbed and interstitial in a SnTe(001) monolayer is comparably low, its local magnetic moment is 0 as a result of the 3d orbitals fully occupied. Considering both the low formation energy and large magnetic moment, Mn is expected to be a prominent choice to introduce magnetism in atomic-thick SnTe in substitutional configuration. Thus, a pristine flat magnetic SnTe monolayer is probable to be obtained.