  Active galaxies, especially blazars, are among the most promising neutrino
source candidates. To date, ANTARES searches for these objects considered
GeV-TeV $\gamma$-ray bright blazars. Here, a statistically complete
radio-bright blazar sample is used as the target for searches of origins of
neutrinos collected by the ANTARES neutrino telescope over 13 years of
operation. The hypothesis of a neutrino-blazar directional correlation is
tested by pair counting and by a complementary likelihood-based approach. The
resulting post-trial $p$-value is $3.0\%$ ($2.2\sigma$ in the two-sided
convention), possibly indicating a correlation. Additionally, a time-dependent
analysis is performed to search for temporal clustering of neutrino candidates
as a mean 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 pre-trial significance above $3\sigma$ indicates
a $p=1.4\%$ ($2.5\sigma$ in the two-sided convention) detection of a
time-variable neutrino flux. An \textit{a posteriori} investigation reveals an
intriguing temporal coincidence of neutrino, radio, and $\gamma$-ray flares of
the J0242+1101 blazar at a $p=0.5\%$ ($2.9\sigma$ 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.
