Within the next few years X-ray free-electron lasers (XFELs) now under construction are expected to generate highly collimated XUV pulses with 1013 photons and a duration of 100 fs. Focusing this radiation to a spot some 10 μm in diameter generates intensities of up to 1016 W/cm2. Such pump intensities make feasible the investigation of photopumped XUV lasers using this radiation. We present simulations taking into account two different mechanisms generating the gain: (1) photoionization with subsequent three-body recombination, which takes advantage of the monochromaticity of the pump radiation to generate very cold electrons; (2) inner-shell ionization in which transient inversion is obtained by generating a hole in an otherwise completely filled shell. The simulations show that under appropriate conditions both mechanisms generate very high gain. However, a number of further issues must be considered, such as the propagation of the pump pulse in the medium to be pumped.