  The object of the present paper is to study $\beta$-almost Yamabe solitons
and $\beta$-almost Ricci solitons on almost co-K\"{a}hler manifolds. In this
paper, we prove that if an almost co-K\"{a}hler manifold $M$ with the Reeb
vector field $\xi$ admits a $\beta$-almost Yamabe solitons with the potential
vector field $\xi$ or $b\xi$, where $b$ is a smooth function then manifold is
$K$-almost co-K\"{a}hler manifold or the soliton is trivial, respectively.
Also, we show if a closed $(\kappa,\mu)$-almost co-K\"{a}hler manifold with
$n>1$ and $\kappa<0$ admits a $\beta$-almost Yamabe soliton then the soliton is
trivial and expanding. Then we study an almost co-K\"{a}hler manifold admits a
$\beta$-almost Yamabe soliton or $\beta$-almost Ricci soliton with $V$ as the
potential vector field, $V$ is a special geometric vector field.
