  The sources of IceCube neutrinos are as yet unknown. The multi-messenger
observation of their emission in $\gamma$-rays can be a guide to their
identification, as exemplified by the case of TXS 0506+056. We suggest a new
method of searching for $\gamma$-rays with Imaging Air Cherenkov Telescopes
from sources in coincidence with possible astrophysical neutrinos. We propose
that searches of $\gamma$-rays are extended, from the current practice of only
a few days, to up to one month from a neutrino alert. We test this strategy on
simulated sources modeled after the blazar \emph{TXS 0506+056-like}, emitting
neutrinos and $\gamma$-rays via photohadronic interactions: the $\gamma$-rays
are subsequently reprocessed in the VHE range. Using MAGIC as a benchmark
example, we show that current Cherenkov Telescopes should be able to
detect$\gamma$-ray counterparts to neutrino alerts with a rate of approximately
one per year. It has been proposed that the high-energy diffuse neutrino flux
can be explained by $\sim$ 5\% of all blazars flaring in neutrinos once every
10 years, with a neutrino luminosity similar to that of TXS 0506+056 during the
2014-2015 neutrino flare. The implementation of our strategy could lead, over a
timescale of one or few years, either to the detection of this subclass of
blazars contributing to the diffuse neutrino flux, or to a constraint on this
model.
