Napoli are also interested in the full-back but the Italian website report Southampton have moved to steal ahead in the battle to sign him.
Malcuit was signed under Marcelo Bielsa at Lille last summer from Saint Etienne.
The 27-year-old began his career with Monaco before dropping down into France's fourth tier to play regularly, and working his way back up to Ligue 1.
Cedric Soares, 26, is Southampton's first choice right-back, and Malcuit would have to compete with him for a starting spot.
If Napoli do push ahead with their own interest in Malcuit, they would appear to have an advantage over Southampton in the race to sign him as they can offer Champions League football.
The last player Southampton signed from Lille was Sofiane Boufal, who has just been loaned to Celta Vigo.
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