The 23-year-old joined Atletico from Villarreal in 2015 but spent last season on loan at Sevilla, where he scored 10 goals in 31 appearances.
Southampton are in the market for a new forward this summer having sold Jay Rodriguez to West Bromwich Albion, leaving Shane Long, Charlie Austin and Manolo Gabbiadini as their only recognised senior striking options.
Sky Sports News reports that new Saints boss Mauricio Pellegrino has identified Vietto as a possible target, with the Argentine below the likes of Antoine Griezmann, Kevin Gameiro and Fernando Torres in the pecking order at Atletico.
However, Southampton could face opposition for the striker's signature from Serie A side Sampdoria, who have also been credited with an interest.
The Saints have made just one signing so far this summer, bringing in Jan Bednarek from Lech Poznan for an undisclosed fee.