Context. The discovery of the unique source HESS J1507-622 in the very high energy (VHE) range (100 GeV-100 TeV) opened new possibilities to study the parent population of ultra-relativistic particles found in astrophysical sources and underlined the possibility of new scenarios/mechanisms crucial for understanding the underlying astrophysical processes in nonthermal sources. Aims. The follow-up X-ray (0.2 - 10 keV) observations on HESS J1507-622 are reported, and possibilities regarding the nature of the VHE source and that of the newly discovered X-ray sources are investigated. Methods.We obtained bservations with the X-ray satellites XMM-Newton and Chandra. Background corrections were applied to the data to search for extended diffuse emission. Since HESS J1507-622 covers a large part of the field of view of these instruments, blank-sky background fields were used. Results. The discovery of several new X-ray sources and a new, faint, extended X-ray source with a flux of ~6e-14 erg cm^-2 s^-1 is reported. Interestingly, a new, variable point-like X-ray source with a flux of ~8e-14 erg cm^-2 s^-1 appeared in the 2011 observation, which was not detected in the previous X-ray observations. Conclusions. The X-ray observations revealed a faint, extended X-ray source that may be a possible counterpart for HESS J1507-622. This source could be an X-ray pulsar wind nebula (PWN) remnant of the larger gamma-ray PWN, which is still bright in IC emission. Several interpretations are proposed to explain the newly detected variable X-ray source.