We investigate close encounters between stellar-mass black holes (BHs) in the gaseous discs of active galactic nuclei (AGNs), during which binary black holes (BBHs) may form. We perform a suite of 483 2D adiabatic viscous hydrodynamic simulations within a shearing box prescription using the Eulerian grid code Athena++. We co-evolve the two embedded BHs with the gas. To probe the dependence of capture on non-circular initial conditions, we vary the initial radial separation b, the eccentricity e of one of the stellar BHs around the central supermassive black hole, and the eccentric phase angle φₑ. We consider eccentricities from e=0 to e=0.1 and compare them with the local disc aspect ratio h=H/R₀≃0.005. We find that small eccentricities shift the capture window in the parameter space. Eccentricities of order the disc aspect ratio, in particular e∼ h--2h, produce successful captures at initial separations that do not capture in the circular models. By contrast, systems with e ≫ h retain less gas before encounter and have lower capture fractions across the sampled parameter grid. We find that the first periapsis distance is a useful predictor of direct capture. Predicting direct capture for r_(rm p)<0.1r_H correctly classifies 92.7% of Hill sphere encounters. The Hill sphere gas mass helps identify gas-poor failures, but provides no clear additional capture boundary within this suite, in which the initial disc density, temperature, and viscosity parameter are held fixed. Pre-encounter eccentricity modifies gas-assisted BBH formation through its combined effects on the first-encounter geometry and the gas reservoir available for orbital energy dissipation.