The recent high-precision measurement of the $W$ mass by the CDF collaboration is in sharp tension with the Standard Model prediction as obtained by the electroweak fit. If confirmed, this finding can only be explained in terms of new physics effects. In this work, we point out a generic connection between the $M_W$ anomaly and Higgs physics observables such as $h\toγγ, Zγ$ and the ratio $h\to ZZ/WW$. Moreover, we systematically classify new physics scenarios which can address the $M_W$ anomaly via a tree-level contribution to the $\hat T$ parameter. These include a real scalar triplet, a scalar quadruplet with the same hypercharge of the Higgs doublet, a $Z'$ boson, a vector triplet with unit hypercharge and a vector boson with the gauge quantum numbers of the Higgs doublet. These solutions to the $M_W$ anomaly are characterized by new physics states which are typically too heavy to be discovered in direct searches, but which might leave their imprints in Higgs physics.

13 pages, 1 figure. Version to appear in Physics Letters B