  Using the idea of quasi-ideals of $P$-regular nearrings, the concept of
bi-ideals of $P$-regular nearrings is generalized, which is an extension of the
concept of quasi-ideals of $P$-regular nearrings and some interesting
characterizations of bi-ideals are obtained. As a result, we prove that every
element of a bi-ideal $B$ of a $P$-regular nearring can be represented as the
sum of two elements of $P$ and $Q$. Moreover, every element of the finite
intersection $\displaystyle \bigcap_{i=1}^{n}B_{i}$ of bi-ideals of a
$P$-regular distributive nearring $N$ can be represented as the sum of two
elements of $P$ and $B_{1}NB_{2}N...NB_{n-1}NB_{n}$.
