  The detection of extended TeV $\gamma$-ray emission (dubbed "TeV halos") from
Geminga and Monogem pulsars by HAWC collaboration implies that the halo-like
morphologies around middle-aged pulsars may be common. The $\gamma$-rays above
10 TeV are thought to arise from inverse Compton (IC) scattering of
relativistic electrons/positrons in the pulsar halos off cosmic microwave
background photons. In the meanwhile, these electrons and positrons can produce
X-ray synchrotron emission in the interstellar magnetic field, resulting in a
diffuse emission in the X-ray band (namely X-ray halos). Here, we study the
prospect of detecting X-ray halos with eROSITA from 10 middle-aged pulsars with
characteristic age larger than tens of thousands of years in the ATNF pulsar
catalog. Assuming a benchmark value (i.e., $B = 3 \rm \, \mu G$) for the
magnetic field, most of the X-ray halos are found to be bright enough to be
detectable by eROSITA in the energy range of 0.5-2 keV during its four-year
all-sky survey. Among these pulsar halos, three are supposed to produce X-ray
fluxes above the eROSITA sensitivity of the first all-sky survey. Given the
good angular resolution and the large field of view, eROSITA is expected to be
able to measure the spatial distribution of the X-ray halos from sub-pc scale
up to tens of pc scale. The intensity profiles of the X-ray halos are very
useful to constrain the magnetic field and the energy-dependence of the
diffusion coefficient in the pulsar halos.
