Southampton sent a scout to the Swiss Super League clash between champions FC Basel and Grasshoppers for an update on a particular target - likely to be Elyounoussi.
The 23-year-old attacker scored 10 goals and assisted 13 others in the league last season and has begun the current campaign in superb form.
He scored a spectacular overhead kick on Basel's 3-1 win over Luzern, although he was also responsible for an own goal later in the game.
Elyounoussi joined Basel last summer from Molde after scoring 42 goals in 92 appearances for the Norwegian club and he has quickly established himself as Basel's star man.
He extended his contract with the Swiss club in July and is now tied to a new four-year agreement.