Measurements of the ZZ and WW final states in the mass range above the 2mZ and 2mW thresholds provide a unique opportunity to measure the off-shell coupling strength of the Higgs boson. This paper presents constraints on the off-shell Higgs boson event yields normalised to the Standard Model prediction (signal strength) in the ZZ → 4ℓ, ZZ → 2ℓ2ν and WW → eνµν final states. The result is based on pp collision data collected by the ATLAS experiment at the LHC, corresponding to an integrated luminosity of 20.3 fb-1 at a collision energy of √s = 8 TeV. Using the CLs method, the observed 95 % confidence level (CL) upper limit on the off-shell signal strength is in the range 5.1-8.6, with an expected range of 6.7-11.0. In each case the range is determined by varying the unknown gg → ZZ and gg → WW background K-factor from higher-order quantum chromodynamics corrections between half and twice the value of the known signal K-factor. Assuming the relevant Higgs boson couplings are independent of the energy scale of the Higgs boson production, a combination with the on-shell measurements yields an observed (expected) 95 %CL upper limit on ΓH/ΓSM H in the range 4.5-7.5 (6.5-11.2) using the same variations of the background K-factor. Assuming that the unknown gg → VV background K-factor is equal to the signal K-factor, this translates into an observed (expected) 95 % CL upper limit on the Higgs boson total width of 22.7 (33.0) MeV.