Generalising the chiral boundary conditions of $R^{1,3}$ gravity for AdS$_{4}$ gravity, we derive chiral locally AdS$_{4}$ solutions in the Newman-Unti gauge consistent with a variational principle whose asymptotic symmetry algebra we show, to be an infinite-dimensional chiral extension of $\mathrm{so}(2,3)$. This symmetry algebra coincides with the chiral ${\mathrm{bms}}_4$ algebra in the flat space limit with the corresponding solutions mapping to the space of gravitational vacua in $R^{1,3}$ gravity. We posit this symmetry algebra as the chiral version of recently discovered Λ-${\mathrm{bms}}_4$ algebra. We propose line integral charges from the bulk AdS$_{4}$ gravity associated with this asymptotic symmetry algebra and show that they obey the semi-classical limit of a $W$-algebra. We derive this $W$-algebra for finite central charge c and level κ using associativity constraints of 2d CFT and find it to be isomorphic to one of quasi-superconformal algebra that existed in the literature.