The 26-year-old, who has worked with Saints boss Mauricio Pellegrino in the past, has been earmarked as the man to solve the team's problems up front.
The coastal club have netted 23 goals in as many Premier League games, and find themselves just one point above the relegation zone.
According to Sky Sports News, Southampton are still in negotiations with Monaco, who value Carrillo at around £15m.
The Argentine, who has been a Monaco player since his 2015 switch from Estudiantes, has started just two of his 22 appearances in Ligue 1 this season.
Since joining Monaco, Carrillo has netted 20 goals.