A search for a pair of neutral, scalar bosons with each decaying into two W bosons is presented using 36.1 fb(-1) of proton-proton collision data at a centre-of-mass energy of 13 TeV recorded with the ATLAS detector at the Large Hadron Collider. This search uses three production models: non-resonant and resonant Higgs boson pair production and resonant production of a pair of heavy scalar particles. Three final states, classified by the number of leptons, are analysed: two same-sign leptons, three leptons, and four leptons. No significant excess over the expected Standard Model backgrounds is observed. An observed (expected) 95% confidence-level upper limit of 160 (120) times the Standard Model prediction of non-resonant Higgs boson pair production cross-section is set from a combined analysis of the three final states. Upper limits are set on the production cross-section times branching ratio of a heavy scalar X decaying into a Higgs boson pair in the mass range of 260 GeV m(X) 500 GeV and the observed (expected) limits range from 9.3 (10) pb to 2.8 (2.6) pb. Upper limits are set on the production cross-section times branching ratio of a heavy scalar X decaying into a pair of heavy scalars S for mass ranges of 280 GeV m(X) 340 GeV and 135 GeV m(S) 165 GeV and the observed (expected) limits range from 2.5 (2.5) pb to 0.16 (0.17) pb.