Pulses of extreme ultraviolet (XUV) light, with wavelengths between 10 and 100 nm, can be used to image and excite ultra-fast phenomena such as the motion of atomic electrons. Here we show that the illumination of plasma with near-critical electron density may be used as a source of relativistically intense XUV radiation, providing the means for novel XUV-pump-XUV-probe experiments in the non-linear regime. We describe how the optimal regime may be reached by tailoring the lasertarget interaction parameters and by the presence of preplasma. Our results indicate that currently available laser facilities are capable of producing XUV pulses with duration ∼ 10 fs, brilliance in excess of 10 23 photons/s/mm 2 /mrad 2 (0.1% bandwidth) and intensity Iλ 2 10 19 Wcm −2 µm 2 .