A search for the narrow structure, X(5568), reported by the D0 Collaboration in the decay sequence X→B_{s}^{0}π^{±}, B_{s}^{0}→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 B_{s}^{0} mesons, ρ_{X}, were determined at 95% confidence level. The results are N(X)<382 and ρ_{X}<0.015 for B_{s}^{0} mesons with transverse momenta above 10 GeV, and N(X)<356 and ρ_{X}<0.016 for transverse momenta above 15 GeV. Limits are also set for potential B_{s}^{0}π^{±} resonances in the mass range 5550 to 5700 MeV.