The decays B+c→J/ψD+s and B+c→J/ψD*+s are studied with the ATLAS detector at the LHC using a dataset corresponding to integrated luminosities of 4.9 and 20.6 fb−1 of pp collisions collected at centre-of-mass energies √s=7 TeV and 8 TeV, respectively. Signal candidates are identified through J/ψ→μ+μ− and D(*)+s→ϕπ+(γ/π0) decays. With a two-dimensional likelihood fit involving the B+c reconstructed invariant mass and an angle between the μ+ and D+s candidate momenta in the muon pair rest frame, the yields of B+c→J/ψD+s and B+c→J/ψD*+s, and the transverse polarisation fraction in B+c→J/ψD*+s decay are measured. The transverse polarisation fraction is determined to be Γ±±(B+c→J/ψD*+s)/Γ(B+c→J/ψD*+s)=0.38±0.23±0.07, and the derived ratio of the branching fractions of the two modes is BB+c→J/ψD*+s/BB+c→J/ψD+s=2.8+1.2−0.8±0.3 , where the first error is statistical and the second is systematic. Finally, a sample of B+c→J/ψπ+ decays is used to derive the ratios of branching fractions BBB+c→J/ψD*+s/BB+c→J/ψπ+=3.8±1.1±0.4±0.2 and BB+c→J/ψD*+s/BB+c→J/ψπ+=10.4±3.1±1.5±0.6, where the third error corresponds to the uncertainty of the branching fraction of D+s→ϕ(K+K−)π+ decay. The available theoretical predictions are generally consistent with the measurement.