A search is presented for the Higgs boson decay to a pair of electrons (e$^+$e$^-$) in proton-proton collisions at $\sqrt{s}$ = 13 TeV. The data set was collected with the CMS experiment at the LHC between 2016 and 2018, corresponding to an integrated luminosity of 138 fb$^{-1}$. The analysis uses event categories targeting Higgs boson production via gluon fusion and vector boson fusion. The observed upper limit on the Higgs boson branching fraction to an electron pair is 3.0 $\times$ 10$^{-4}$ (3.0 $\times$ 10$^{-4}$ expected) at the 95% confidence level, which is the most stringent limit on this branching fraction to date.