Type II-P supernov AE (SNe), the most common core-collapse SNe type, result from the explosions of red supergiant stars. Their detection in the radio domain testifies of the presence of relativistic electrons, and shows that they are potentially efficient energetic particle accelerators. If hadrons can also be accelerated, these energetic particles are expected to interact with the surrounding medium to produce a gamma-ray signal even in the multi-TeV range. The intensity of this signal depends on various factors, but an essential one is the density of the circumstellar medium. Such a signal should however be limited by electron-positron pair production arising from the interaction of the gamma-ray photons with optical photons emitted by the supernova photosphere, which can potentially degrade the gamma-ray signal by over ten orders of magnitude in the first days/weeks following the explosion. We calculate the gamma-gamma opacity from a detailed modelling of the time evolution of the forward shock and supernova photosphere, taking a full account of the non-isotropy of the photon interactions. We discuss the time-dependent gamma-ray TeV emission from Type II-P SNe as a function of the stellar progenitor radius and mass-loss rate, as well as the explosion energy and mass of the ejected material. We evaluate the detectability of the SNe with the next generation of Cherenkov telescopes. We find that, while most extragalactic events may be undetectable, Type II-P SNe exploding in our Galaxy or in the Magellanic Clouds should be detected by gamma-ray observatories such as the upcoming Cherenkov Telescope Array.