The phenomenon of jetting and accompanying droplet ejection is known to occur in many fluid dynamical situations involving collapse of a gaseous cavity at a liquid-gas interface (Veron, Ann. Rev. Fluid Mech. 47, 2015). In a recent study of free, capillary-gravity oscillations on a quiescent cylindrical pool of liquid (Farsoiya et. al., Journal of Fluid Mechanics 826, 2017), it has been shown using Direct Numerical Simulations (DNS) that jetting may be obtained with an initial interfacial perturbation taken to be a Bessel mode (primary mode) of the form η(r, 0) = a0 J0 (lp r / R0) with R0 being the radius of the domain (lp denotes the pth non-trivial root of J1(r)). For sufficiently large values of epsilon ≡ a0 lp / R0, it was found that waves at the interface produce a jet at the axis of symmetry which can eject droplets from its tip. This jetting is a nonlinear phenomena involving the generation of higher modes and energy transfer to these. We develop a weakly nonlinear, irrotational, inviscid theory which is able to capture the initiation of jetting.