The recent numerical study of Barthelemy et al. (J. Fluid Mech., vol. 841, 2018, pp. 463–488) investigated the local properties of two-dimensional (2-D) and three-dimensional (3-D) nonlinear unsteady gravity wave packets in deep and uniform intermediate depth water. Their study focused on the breaking inception transition zone separating maximum recurrence and marginal breaking, and reported that a suitably normalized energy flux localized at the steepest crest in the packet provides a robust breaking threshold parameter. Our present study uses the fully nonlinear boundary integral element method solver developed by Touboul & Kharif (Nat. Haz., vol. 84, issue 2, 2016, pp. 585–598) to investigate breaking inception of 2-D deep water nonlinear water wave packets propagating in the presence of a background current that varies linearly with depth. We seek to validate whether the proposed generic breaking inception threshold holds for the case of constant background vorticity. Results are presented for different packet bandwidths and background vorticity levels.