The B0→K∗0K¯¯¯¯¯∗0 and B0s→K∗0K¯¯¯¯¯∗0 decays are studied using proton-proton collision data corresponding to an integrated luminosity of 3 fb−1. An untagged and timeintegrated amplitude analysis of B ( s) 0  → (K+π−)(K−π+) decays in two-body invariant mass regions of 150 MeV/c2 around the K∗0 mass is performed. A stronger longitudinal polarisation fraction in the B0→K∗0K¯¯¯¯¯∗0 decay, fL = 0.724 ± 0.051 (stat) ± 0.016 (syst), is observed as compared to fL = 0.240 ± 0.031 (stat) ± 0.025 (syst) in the B0s→K∗0K¯¯¯¯¯∗0 decay. The ratio of branching fractions of the two decays is measured and used to determine B(B0→K∗0K¯¯¯¯¯∗0)=(8.0±0.9(stat)±0.4(syst))×10−7 .