The two Premier League clubs have had proposals in the region of £22m plus add-ons accepted by the Canaries.
And one source has told Sky Sports News that Maddison’s preferred choice is to sign for Southampton.
He has also been scouted by top-four clubs Tottenham, Liverpool and Manchester City.
The 21-year-old scored 15 goals last season and, with their parachute payments due to end this year, Norwich have been willing to cash in on their most valuable player.
The Norfolk club have been seeking between £20m and 25m for Maddison, who has represented England at U21 level.