The 20-year-old impressed on loan at the Red Bull Arena last season, scoring five goals and registering three assists in 11 appearances.
Lookman's exploits helped Leipzig finish sixth in the Bundesliga table and secure a Europa League second qualifying round place.
According to The Mirror, Red Bull have held initial talks with Everton about signing the England Under-21 international on a long-term basis.
The report goes on to claim that Lookman's turn in form has also caught the eye of both Spurs and Saints, though the Toffees are thought to be reluctant to let the player leave permanently.
Lookman joined Everton from Charlton Athletic in January 2017 and is contracted to the Goodison Park outfit until June 2021.