The 18-year-old, who won the club's Academy Player of the Award last season, has asked for a transfer.
The Daily Mail says Saints have agreed a £5m deal for the highly rated defender.
Livramento will then sign a five-year contract with his new club.
It is the second time Southampton have raided their Premier League rivals this summer, having recently signed centre-back Dynel Simeu.
Brighton and AC Milan had also pursued Livramento.
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