In this paper we present a theoretical study of the structure, energetics, potential energy surfaces, and energetic stability of excess electron bubbles in (He4) N (N=6500- 106) clusters. The subsystem of the helium atoms was treated by the density functional method. The density profile was specified by a void (i.e., an empty bubble) at the cluster center, a rising profile towards a constant interior value (described by a power exponential), and a decreasing profile near the cluster surface (described in terms of a Gudermannian function). The cluster surface density profile width (∼6 Å) weakly depends on the bubble radius Rb, while the interior surface profile widths (∼4-8 Å) increase with increasing Rb. The cluster deformation energy Ed accompanying the bubble formation originates from the bubble surface energy, the exterior cluster surface energy change, and the energy increase due to intracluster density changes, with the latter term providing the dominant contribution for N=6500-2× 105. The excess electron energy Ee was calculated at a fixed nuclear configuration using a pseudopotential method, with an effective (nonlocal) potential, which incorporates repulsion and polarization effects. Concurrently, the energy V0 of the quasi-free-electron within the deformed cluster was calculated. The total electron bubble energies Et = Ee + Ed, which represent the energetic configurational diagrams of Et vs Rb (at fixed N), provide the equilibrium bubble radii Rbc and the corresponding total equilibrium energies Ete, with Ete (Re) decreasing (increasing) with increasing N (i.e., at N=6500, Re =13.5 Å and Ete =0.86 eV, while at N=1.8× 105, Re =16.6 Å and Ete =0.39 eV). The cluster size dependence of the energy gap (V0 - Ete) allows for the estimate of the minimal (He4) N cluster size of N≃5200 for which the electron bubble is energetically stable.