  We study the properties of the central spheroids located within 10 kpc of the
centre of mass of Milky Way mass-sized galaxies simulated in a cosmological
context. The simulated central regions are dominated by stars older than 10
Gyr, mostly formed in situ, with a contribution of ~30 per cent from accreted
stars. These stars formed in well-defined starbursts, although accreted stars
exhibit sharper and earlier ones. The fraction of accreted stars increases with
galactocentric distance, so that at a radius of ~8-10 kpc a fraction of ~40 per
cent, on average, are detected. Accreted stars are slightly younger, lower
metallicity, and more $\alpha$-enhanced than in situ stars. A significant
fraction of old stars in the central regions come from a few ($2-3$) massive
satellites ($\sim 10^{10}{\rm M}_\odot$). The bulge components receive larger
contributions of accreted stars formed in dwarfs smaller than $\sim
10^{9.5}{\rm M}_\odot$. The difference between the distributions of ages and
metallicities of old stars is thus linked to the accretion histories -- those
central regions with a larger fraction of accreted stars are those with
contributions from more massive satellites. The kinematical properties of in
situ and accreted stars are consistent with the latter being supported by their
velocity dispersions, while the former exhibit clear signatures of rotational
support. Our simulations demonstrate a range of characteristics, with some
systems exhibiting a co-existing bar and spheroid in their central regions,
resembling in some respect the central region of the Milky Way.
