  The cosmological lithium problem, that is, the discrepancy between the
lithium abundance predicted by the Big Bang nucleosynthesis and the one
observed for the stars of the 'Spite plateau', is one of the long standing
problems of modern astrophysics. Recent hints for a possible solution involve
lithium burning induced by protostellar mass accretion on Spite plateau stars.
  The purpose of this paper is to analyze the effect of protostellar accretion
on low metallicity low-mass stars with a focus on PMS lithium evolution. We
computed the evolution from the protostar to the MS phase of accreting models
with final masses of 0.7 and 0.8 M$_\odot$, and three metallicities Z=0.0001,
Z=0.0010, and Z=0.0050. The effects of changing the main parameters affecting
accreting models, that is the accretion energy (cold versus hot accretion), the
initial seed mass $M_{seed}$ and radius $R_{seed}$, and the mass accretion rate
$\dot{m}$, have been investigated in detail.
  As for the main stellar properties and the surface $^7 Li$ abundance, hot
accretion models converge to standard non-accreting ones within 1 Myr,
regardless of the actual value of $M_{seed}$, $R_{seed}$, and $\dot{m}$. Also,
cold accretion models with a relatively large $M_{seed}$ ($\gtrsim 10~M_{jup}$)
or $R_{seed}$ ($\gtrsim 1~R_\odot$) converge to standard non-accreting ones in
less than about 10-20~Myr. A drastically different evolution occurs whenever a
cold protostellar accretion process starts from small values of $M_{seed}$ and
$R_{seed}$ ($M_{seed}\sim 1~M_{jup}$, $R_{seed} \lesssim 1~R_\odot$). These
models almost entirely skip the standard Hayashi track evolution and deplete Li
before the end of the accretion phase. The exact amount of depletion depends on
the actual combination of the accretion parameters ($\dot{m}$, $M_{seed}$, and
$R_{seed}$), achieving in some cases the complete exhaustion of Li in the whole
star.
