Context. The source(s) of the neutrino excess reported by the IceCube Collaboration is unknown. The TANAMI Collaboration recently reported on the multiwavelength emission of six bright, variable blazars which are positionally coincident with two of the most energetic IceCube events. Objects like these are prime candidates to be the source of the highest-energy cosmic rays, and thus of associated neutrino emission. Aims. We present an analysis of neutrino emission from the six blazars using observations with the ANTARES neutrino telescope. Methods. The standard methods of the ANTARES candidate list search are applied to six years of data to search for an excess of muons - and hence their neutrino progenitors - from the directions of the six blazars described by the TANAMI Collaboration, and which are possibly associated with two IceCube events. Monte Carlo simulations of the detector response to both signal and background particle fluxes are used to estimate the sensitivity of this analysis for different possible source neutrino spectra. A maximum-likelihood approach, using the reconstructed energies and arrival directions of through-going muons, is used to identify events with properties consistent with a blazar origin. Results. Both blazars predicted to be the most neutrino-bright in the TANAMI sample (1653-329 and 1714-336) have a signal flux fitted by the likelihood analysis corresponding to approximately one event. This observation is consistent with the blazar-origin hypothesis of the IceCube event IC14 for a broad range of blazar spectra, although an atmospheric origin cannot be excluded. No ANTARES events are observed from any of the other four blazars, including the three associated with IceCube event IC20. This excludes at a 90% confidence level the possibility that this event was produced by these blazars unless the neutrino spectrum is flatter than -2.4.