The Independent reports that Lookman, 21, is available for transfer after making just three league starts for the Blues this season, and could depart for a £20 million fee.
A move to St Mary’s Stadium would see Lookman reunite with Saints manager Ralph Hasenhüttl, who was in charge of RB Leipzig during the youngster’s loan spell at the Bundesliga club during the second half of last season.
The German outfit attempted to sign Lookman permanently last summer, but did not meet Everton’s reported valuation of £28 million.
Leipzig are said to still be interested, as are Eddie Howe’s Bournemouth.
Lookman left his home town club Charlton for Goodison Park in January 2017 for roughly £11 million, and scored on his debut in a 4-0 win over Manchester City.
But he has not managed a single league goal since then, and has made just 48 Blues appearances in two-and-a-half years.
Lookman, who Everton boss Marco Silva described as the club’s ‘present and future’ before the start of this season, has two years left on his current contract at Goodison.