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Superconductivity can be induced in a normal material via the ‘leakage’ of superconducting pairs of charge carriers from an adjacent superconductor. This so-called proximity effect is markedly influenced by graphene’s unique electronic structure, both in fundamental and technologically relevant ways. These include an unconventional form of the ‘leakage’ mechanism—the Andreev reflection—and the potential of supercurrent modulation through electrical gating. Despite the interest of high-temperature superconductors in that context, realizations have been exclusively based on low-temperature ones. Here we demonstrate a gate-tunable, high-temperature superconducting proximity effect in graphene. Notably, gating effects result from the perfect transmission of superconducting pairs across an energy barrier—a form of Klein tunnelling, up to now observed only for non-superconducting carriers—and quantum interferences controlled by graphene doping. Interestingly, we find that this type of interference becomes dominant without the need of ultraclean graphene, in stark contrast to the case of low-temperature superconductors. These results pave the way to a new class of tunable, high-temperature Josephson devices based on large-scale graphene.