Prospects for the study of Higgs boson pair (HH) production in the $\mathrm{HH \to b\bar{b}4l}$ (l = e, $\mu$) channel are studied in the context of the High Luminosity LHC. The analysis is performed using a parametric simulation of the Phase-2 CMS detector response provided by the Delphes software and assuming an average of 200 proton-proton collisions per bunch crossing at a center-of-mass energy of 14 TeV. Assuming a projected integrated luminosity of 3000 fb$^{-1}$, the expected significance for the nonresonant standard model (SM) HH signal is 0.37 $\sigma$; a 95$\%$ confidence level (CL) upper limit on its cross section is set to 6.6 times the SM prediction. The statistical combination of five decay channels ($\mathrm{b\bar{b}b\bar{b}}$, $\mathrm{b\bar{b}\tau\tau}$, $\mathrm{b\bar{b}\gamma\gamma}$, $\mathrm{b\bar{b}WW}$, $\mathrm{b\bar{b}ZZ}$) results in an expected significance for the SM HH signal of 2.6 $\sigma$ and an expected 68$\%$ and 95$\%$ CL intervals for the self-coupling modifier $\kappa_{\lambda}$ = $\lambda_{HHH}/\lambda_{HHH}^{SM}$ of [$0.35, 1.9$] and [$-0.18, 3.6$], respectively.