Active galaxies, especially blazars, are among the most promising extragalactic candidates for high-energy neutrino sources. To date, ANTARES searches included these objects and used GeV–TeV γ-ray flux to select blazars. Here, a statistically complete blazar sample selected by their bright radio emission is used as the target for searches of origins of neutrinos collected by the ANTARES neutrino telescope over 13 yr of operation. The hypothesis of a neutrino–blazar directional correlation is tested by pair counting and a complementary likelihood-based approach. The resulting posttrial p-value is 3.0% (2.2σ in the two-sided convention). Additionally, a time-dependent analysis is performed to search for temporal clustering of neutrino candidates as a means of detecting neutrino flares in blazars. None of the investigated sources alone reaches a significant flare detection level. However, the presence of 18 sources with a pretrial significance above 3σ indicates a p = 1.4% (2.5σ in the two-sided convention) detection of a time-variable neutrino flux. An a posteriori investigation reveals an intriguing temporal coincidence of neutrino, radio, and γ-ray flares of the J0242+1101 blazar at a p = 0.5% (2.9σ in the two-sided convention) level. Altogether, the results presented here suggest a possible connection of neutrino candidates detected by the ANTARES telescope with radio-bright blazars.