  We present the results of the analysis of deep archival \sat\ observations
towards the dwarf spheroidal galaxies Draco, Leo I, Ursa Major II and Ursa
Minor in the Milky Way neighbourhood. The X-ray source population is
characterized and cross-correlated with available databases with the aim to
infer their nature. We also investigate if intermediate-mass black holes are
hosted in the center of these galaxies. In the case of Draco, we detect 96
high-energy sources, two of them being possibly local stars, while no evidence
for any X-ray emitting central compact object is found. Towards the Leo I and
UMa II field of view we reveal 116 and 49 X-ray sources, respectively. None of
them correlates with the putative central black holes and only one is likely
associated with a UMa II local source. The study of the UMi dwarf galaxy shows
54 high-energy sources and a possible association {with a source at the dSph
center}. We put an upper limit to the central compact object luminosity of
4.02$\times$10$^{33}$ erg/s. Furthermore, via the correlation with a radio
source near the galactic center, we get that the putative black hole should
have a mass of $\left(2.76^{+32.00}_{-2.54}\right)\times10^6 M_{\odot}$ and be
radiatively inefficient. This confirms a previous result obtained by using
Chandra data alone.
