Using angle-resolved photoemission spectroscopy, we observe the low-temperature state of the A_{x}Fe_{2-y}Se_{2} (A=K, Rb) superconductors to exhibit an orbital-dependent renormalization of the bands near the Fermi level-the d_{xy} bands heavily renormalized compared to the d_{xz}/d_{yz} bands. Upon raising the temperature to above 150 K, the system evolves into a state in which the d_{xy} bands have depleted spectral weight while the d_{xz}/d_{yz} bands remain metallic. Combined with theoretical calculations, our observations can be consistently understood as a temperature-induced crossover from a metallic state at low temperatures to an orbital-selective Mott phase at high temperatures. Moreover, the fact that the superconducting state of A_{x}Fe_{2-y}Se_{2} is near the boundary of such an orbital-selective Mott phase constrains the system to have sufficiently strong on-site Coulomb interactions and Hund's coupling, highlighting the nontrivial role of electron correlation in this family of iron-based superconductors.