  We study those rings in which all invertible elements are weakly nil-clean
calling them {\it UWNC rings}. This somewhat extends results due to
Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all
invertible elements are nil-clean were considered abbreviating them as {\it UNC
rings}. Specifically, our main achievements are that the triangular matrix ring
${\rm T}_n(R)$ over a ring $R$ is UWNC precisely when $R$ is UNC. Besides, the
notions UWNC and UNC do coincide when $2 \in J(R)$. We also describe UWNC
$2$-primal rings $R$ by proving that $R$ is a ring with $J(R) = {\rm Nil}(R)$
such that $U(R)=\pm 1+{\rm Nil}(R)$. In particular, the polynomial ring $R[x]$
over some arbitrary variable $x$ is UWNC exactly when $R$ is UWNC. Some other
relevant assertions are proved in the present direction as well.
