Searches for the Higgs boson decays H→ee and H→eμ are performed using data corresponding to an integrated luminosity of 139fb collected with the ATLAS detector in pp collisions at s=13 TeV at the LHC. No significant signals are observed, in agreement with the Standard Model expectation. For a Higgs boson mass of 125 GeV, the observed (expected) upper limit at the 95% confidence level on the branching fraction B(H→ee) is 3.6×10 (3.5×10) and on B(H→eμ) is 6.2×10 (5.9×10). These results represent improvements by factors of about five and six on the previous best limits on B(H→ee) and B(H→eμ) respectively.