We find an exact spherically symmetric regular Bardeen-like solutions by considering the coupling between Einstein-Gauss-Bonnet theory and nonlinear electrodynamics (NED) in five-dimensional spacetime. These solutions, with an additional parameter g apart from the mass M , represent black holes with Cauchy and event horizons, extremal black holes with degenerate horizons or no black holes in the absence of the horizons, and encompasses as a special case Boulware-Deser black holes which can be recovered in the absence of magnetic charge (g = 0). Owing to the NED corrected black hole, the thermodynamic quantities have also been modified and we have obtained exact analytical expressions for the thermodynamical quantities such the Hawking temperature T + , the entropy S + , the specific heat C + , and the Gibbs free energy F + . The heat capacity diverges at a critical radius r = r C , where incidentally the temperature has a maximum, and the Hawking-Page transitions even in absence of the cosmological term. The thermal evaporation process leads to eternal remnants for sufficiently small black holes and evaporates to a thermodynamic stable extremel black hole remnants with vanishing temperature. The heat capacity becomes positive C + > 0 for r + < r C allowing black hole to become thermodynamically stable, in addition the smaller black holes are globally stable with positive heat capacity C + > 0 and negative free energy F + < 0 . The entropy S of a 5D Bardeen black hole is not longer a quarter of the horizons area A, i.e., S = A/4