A search for the narrow structure, X(5568), reported by the D0 Collaboration in the decay sequence X → Bₛ⁰π±, Bₛ⁰ → J/ψϕ, is presented. The analysis is based on a data sample recorded with the ATLAS detector at the LHC corresponding to 4.9 fb⁻¹ of pp collisions at 7 TeV and 19.5 fb⁻¹ at 8 TeV. No significant signal was found. Upper limits on the number of signal events, with properties corresponding to those reported by D0, and on the X production rate relative to Bₛ⁰ mesons, ρᵪ, were determined at 95% confidence level. The results are N(X) < 382 and ρᵪ < 0.015 for Bₛ⁰ mesons with transverse momenta above 10 GeV, and N(X) < 356 and ρᵪ < 0.016 for transverse momenta above 15 GeV. Limits are also set for potential Bₛ⁰π± resonances in the mass range 5550 to 5700 MeV.