Compact 3-dimensional extension of two-dimensional torus

Question

A two-dimensional torus $T^2=S^1\times S^1$ can be extended to a solid torus or so-called toroid which is a topological space equivalent with $D^2\times S^1$. However, such an extension does not treat the two generating cycles of $T^2$ equivalently. For example, one of them will be the boundary of $D^2$ thereby contractible, while the other one is non-contractible. A direct negative result is that not all smooth function on $T^2$ can be extended to this three-dimensional toroid. Therefore, is there any three-dimensional compact manifold $M$ with $\partial M=T^2$, where all the smooth function $T^2\rightarrow\mathbb{R}/\mathbb{Z}$ on its boundary $T^2$ can be extended into the ''bulk'' $M$?