The atmospheric convection is a phenomenon that has a length scale smaller than the resolution at which the state-of-the-art general circulation numerical models are running, and thus, the convection needs to be parameterized. In this work, a theoretical formulation for the parameterization of subgrid-scale convection based on subgrid averaging that represents a generalized eddy-diffusivity mass-flux (GEM) formulation is presented. The subgrid fluxes are derived by considering a decomposition of subgrid variables into convective and turbulent variables, and by assuming that the convection is modeled by round convective plumes with generic radial profiles. The main difference between our formulation and the mass-flux formulations is that the condition of a very small fractional area occupied by the convection is replaced with the condition that the convective vertical velocity goes to the mean state far from the updraft region of the plume. The plume model can be also generalized by considering that the convective elements are energy-consistent plumes, governed by the conservation of mass, momentum, kinetic energy, and buoyancy, which provides a physical-based closure for the entrainment. Therefore, the closing problem of our formulation consists of the specification of the convective radial profiles and the boundary conditions at the initial vertical level. Furthermore, our formulation allows one to consider more realistic profiles for the convective variables, making the formulation suitable for the parameterization of atmospheric convection at the convective gray-zone. This aspect is discussed in the last part of this work, where it is showed how the formulation can be implemented at any resolution. Moreover, in the present framework, no distinct assumptions are required for each convective type, thus facilitating the parameterization of convection in a unified way.