We construct embedded Willmore tori with small area constra int in Riemannian three-manifolds under some curvature condition used to prevent M ̈obius dege neration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new ge ometric expansions of exponentiated small symmetric Clifford tori and analyze the sharp asymptot ic behavior of degenerating tori under the action of the M ̈obius group. In this first work we prove two existence results by minimizing or maximizing a suitable reduced functional, in particular we obtain embedded area-constrained Willmore tori (or, equivalently, toroidal critical points of the Hawking mass under area-constraint) in compact 3-manifolds with constant scalar curvature and i n the double Schwarzschild space. In a forthcoming paper new existence theorems will be achieved v ia Morse theory.