  Let $G$ be a connected reductive algebraic group. Let $\mathcal{E}\rightarrow
\mathcal{B}$ be a principal $G\times G$-bundle and $X$ be a regular
compactification of $G$. We describe the Grothendieck ring of the associated
fibre bundle $\mathcal{E}(X):=\mathcal{E}\times_{G\times G} X$, as an algebra
over the Grothendieck ring of a canonical toric bundle over a flag bundle on
$\mathcal{B}$. These are relative versions of the results on equivariant
$K$-theory of regular compactifications of $G$. They also generalize the well
known results on the Grothendieck rings of projective bundles, toric bundles
and flag bundles.
