3EG J1837-0423 and HESS J1841-055 are two unidentified and peculiar high-energy sources located in the same region of the sky, separated by ~1.4°. Specifically, 3EG J1837-0423 is a transient MeV object detected by EGRET only once during flaring activity that lasted a few days while HESS J1841-055 is a highly extended TeV source. We attempted to match the high-energy emission from the unidentified sources 3EG J1837-0423 and HESS J1841-055 with X-rays (4-20 keV) and soft γ-rays (20-100 keV) candidate counterparts detected through deep International Gamma-Ray Astrophysics Laboratory observations of the sky region. As a result we propose the Supergiant Fast X-ray Transient (SFXT) AX J1841.0-0536 as a possible candidate counterpart of 3EG J1837-0423, based on spatial proximity and transient behavior. Alternatively, AX J1841.0-0536 could be responsible for at least a fraction of the entire TeV emission from the extended source HESS J1841-055, based on a striking spatial correlation. In either case, the proposed association is also supported from an energetic standpoint by a theoretical scenario where AX J1841.0-0536 is a low magnetized pulsar which, due to accretion of massive clumps from the supergiant companion donor star, undergoes sporadic changes to transient Atoll-states where a magnetic tower can produce transient jets and as a consequence high-energy emission. In either case (by association with 3EG J1837-0423 or alternatively with HESS J1841-055), AX J1841.0-0536 might be the prototype of a new class of Galactic transient MeV/TeV emitters.