The 23-year-old left-back has emerged as a target for Jurgen Klopp in the dying embers of this transfer window.
According to Gianluca Di Marzio, Liverpool are in talks with Roma over a deal.
Palmieri - currently sidelined with a knee injury - reportedly fancies the switch to Anfield.
Despite being out of action until October, Liverpool are set to table an offer.
Italian publication TuttoMercatoWeb claim a £19.3m (€21m) bid is ready to be launched.
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Liverpool have struggled at left-back in recent seasons.
Alberto Moreno has never impressed so midfielder James Milner filled in last term.
Scotland international Andrew Robertson was snapped up from Hull earlier this summer for £8m.
He is relatively inexperienced at the top level and Klopp is eager to beef up his squad ahead of Liverpool’s Champions League adventure.
Palmieri, who can also play on the left-hand side of midfield, has made 45 appearances for Roma.
He is under contract with the Italian capital club until 2021.
