on behalf of the ANTARES CollaborationThe principles of a search for high energy neutrino emission in coincidence with very high energy (VHE; 0.1-100 TeV) gamma-ray flares from two bright extragalactic sources, based on the data collected in 2012-2016 by the ANTARES neutrino detector and High Altitude Water Cherenkov (HAWC) gamma-ray observatory, are presented. The ANTARES telescope observes with high duty cycle an instantaneous field of view of 2π. To study variable sources like blazars, it is crucial to achieve unbiased monitoring and an instrument like HAWC is capable of long term and continuous monitoring of the source with nearly 100% duty cycle. Markarian 421 (Mrk 421) and Markarian 501 (Mrk 501) are the brightest and the closest BL Lac objects known. In contrast to other types of active galactic nuclei (AGN), BL Lacs are characterized by rapid and largeamplitude flux variability. Such radio-loud AGNs with collimated jets aligned to the line of sight, are candidate sources of the observed high energy cosmic rays and of accompanying neutrinos and gamma rays produced in hadronic interactions with the surrounding medium.