  Blazar flares have been suggested as ideal candidates for enhanced neutrino
production. While the neutrino signal of $\gamma$-ray flares has been widely
discussed, the neutrino yield of X-ray flares has received less attention.
Here, we compute the predicted neutrino signal from X-ray flares detected in 66
blazars observed more than 50 times with the X-ray Telescope (XRT) on board the
Neil Gehrels Swift Observatory. We consider a scenario where X-ray flares are
powered by synchrotron radiation of relativistic protons, and neutrinos are
produced through photomeson interactions between protons with their own
synchrotron X-ray photons. Using the 1 keV X-ray light curves for flare
identification, the 0.5-10 keV fluence of each flare as a proxy for the
all-flavour neutrino fluence, and the IceCube point-source effective area for
different detector configurations, we calculate the number of muon and antimuon
neutrinos above 100 TeV expected for IceCube from each flaring source. The bulk
of the neutrino events from the sample originates from flares with durations
$\sim 1-10$ d. Accounting for the X-ray flare duty cycle of the sources in the
sample, which ranges between $\sim2$ and 24 per cent, we compute an average
yearly neutrino rate for each source. The median of the distribution (in
logarithm) is $\sim0.03$ yr$^{-1}$, with Mkn 421 having the highest predicted
rate $1.2\pm 0.3$ yr$^{-1}$, followed by 3C 273 $(0.33\pm0.03$ yr$^{-1})$ and
PG 1553+113 ($0.25\pm0.02$ yr$^{-1}$). Next-generation neutrino detectors
together with regular X-ray monitoring of blazars could constrain the duty
cycle of hadronic X-ray flares.
