The bid, which would rank only behind summer arrival Sofiane Boufal for the biggest in the club's history and consists of €16m (£13.6m) plus a further €4m (£3.4m) dependent on appearances made in the Premier League.
Southampton manager Claude Puel is looking to bolster his forward line, which Shane Long netting just twice this season and Charlie Austin sidelined with a shoulder injury until April.
Gabbiadini has scored five times for Napoli this term after being restricted to just 10 starts across on competitions.
Southampton have struggled for goals under Puel, netting just 22 times in 22 league outings, with the sidelined Austin their top scorer with six while the ever-improving Jay Rodriguez has pitched in with four.
Gabbiadini, who has scored in all three of his previous games for Napoli, has struggled for game time with Lorenzo Insigne, Dries Mertens and Jose Callejon also available and big summer signing Arkadiusz Milik returning from injury.
There had reportedly been tension in the Napoli camp after president Aurelio De Laurentiis refused to lower his asking price for the 25-year-old, with suitors unprepared to meet the valuation.
Southampton are believed to be facing competition from Premier League rivals West Brom, West Ham and Watford though are understood to be leading the chase for the six-time Italy international's signature.