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, ρX, were determined at 95% confidence level. The results are N(X)<382 and ρX<0.015 for Bs0 mesons with transverse momenta above 10 GeV, and N(X)<356 and ρX<0.016 for transverse momenta above 15 GeV. Limits are also set for potential Bs0π± resonances in the mass range 5550 to 5700 MeV.