While there are lots of contributions on Willmore surfaces in the three-dimensional Euclidean space, the literature on curved manifolds is still relatively limited. One of the main aspects of the Willmore problem is the loss of compactness under conformal transformations. We construct embedded Willmore tori in manifolds with a small area constraint by analysing how the Willmore energy under the action of the Möbius group is affected by the curvature of the ambient manifold. The loss of compactness is then taken care of using minimisation arguments or Morse theory.