  The precision measurement of the W boson mass is an important milestone for
the LHC physics program in the coming years. An accurate measurement of $M_W$
allows to perform stringent consistency tests of the Standard Model by means of
global electroweak fits, which in turn are sensitive to New Physics at scales
potentially higher than the ones explored in direct searches. From the
theoretical point of view, our limited knowledge of PDFs will be one of the
dominant sources of uncertainty in ongoing and future LHC determinations of
$M_W$. In this whitepaper, we have quantified the impact of PDF uncertainties
in the W mass extractions from the transverse mass distribution at the LHC. The
calculation has been performed using the NNPDF2.3 set, which includes direct
constrains on the W boson production kinematics with data for electroweak gauge
boson production from the LHC. Our results confirm previous estimates that PDF
uncertainties in the determination of $M_W$ from the $m_W^T$ distribution are
moderate, around 10 MeV at most. We briefly discuss also the case of the lepton
$p_T$ distribution.
