A natural $T$-function on a natural bundle $F$ is a natural operator transforming vector fields on a manifold $M$ into functions on $FM$. For any Weil algebra $A$ satisfying $\dim M \ge {\mathrm width}(A)+1$ we determine all natural $T$-functions on $T^*T^AM$, the cotangent bundle to a Weil bundle $T^AM$.