We have explored the Eu production in the Milky Way by means of a very detailed chemical evolution model. In particular, we have assumed that Eu is formed in merging neutron star (or neutron star black hole) binaries as well as in type II supernovae. We have tested the effects of several important parameters influencing the production of Eu during the merging of two neutron stars, such as: i) the time scale of coalescence, ii) the Eu yields and iii) the range of initial masses for the progenitors of the neutron stars. The yields of Eu from type II supernovae are very uncertain, more than those from coalescing neutron stars, so we have explored several possibilities. We have compared our model results with the observed rate of coalescence of neutron stars, the solar Eu abundance, the [Eu/Fe] versus [Fe/H] relation in the solar vicinity and the [Eu/H] gradient along the Galactic disc. Our main results can be summarized as follows: i) neutron star mergers can be entirely responsible for the production of Eu in the Galaxy if the coalescence time scale is no longer than 1 Myr for the bulk of binary systems, the Eu yield is around $3 \times 10^{-7}$ M$_⊙$, and the mass range of progenitors of neutron stars is 9-50 M$_⊙$; ii) both type II supernovae and merging neutron stars can produce the right amount of Eu if the neutron star mergers produce $2 \times 10^{-7}$ M$_⊙$ per system and type II supernovae, with progenitors in the range 20-50 M$_⊙$, produce yields of Eu of the order of $10^{-8}-10^{-9}$ M$_⊙$; iii) either models with only neutron stars producing Eu or mixed ones can reproduce the observed Eu abundance gradient along the Galactic disc. ; Comment: 9 pages, 8 figures, accepted for publication in MNRAS