Aims. Strong lensing by massive galaxy clusters can provide magnification of the flux and even multiple images of the galaxies that lie behind them. This phenomenon facilitates observations of high-redshift supernovae (SNe) that would otherwise remain undetected. Type Ia supernovae (SNe Ia) detections are of particular interest because of their standard brightness, since they can be used to improve either cluster lensing models or cosmological parameter measurements.Methods. We present a ground-based, near-infrared search for lensed SNe behind the galaxy cluster Abell 370. Our survey was based on 15 epochs of J-band observations with the HAWK-I instrument on the Very Large Telescope (VLT). We use Hubble Space Telescope (HST) photometry to infer the global properties of the multiply-imaged galaxies. Using a recently published lensing model of Abell 370, we also present the predicted magnifications and time delays between the images.Results. In our survey, we did not discover any live SNe from the 13 lensed galaxies with 47 multiple images behind Abell 370. This is consistent with the expectation of 0.09 ± 0.02 SNe calculated based on the measured star formation rate. We compare the expectations of discovering strongly lensed SNe in our survey and that performed with HST during the Hubble Frontier Fields (HFF) programme. We also show the expectations of search campaigns that can be conducted with future facilities, such as the James Webb Space Telescope (JWST) or the Wide-Field Infrared Survey Telescope (WFIRST). We show that the NIRCam instrument aboard the JWST will be sensitive to most SN multiple images in the strongly lensed galaxies and thus will be able to measure their time delays if observations are scheduled accordingly.Key words: gravitational lensing: strong / supernovae: general / galaxies: clusters: individual: A 370⋆ Based on observations made with European Southern Observatory (ESO) telescopes at the Paranal Observatory under programme ID 091.A-0108 and ID 093.A-0278 PI: A. Goobar.