Swiss newspaper Blick claims that on Saturday night the Southampton scout was "zealously recording his observations whenever Elyounoussi was on the ball" and that "at the end of the game, his notebook was full".
The 23-year-old Norwegian, who plays on the left of a three-man forward line for the Swiss champions, has been on Saints radar for the past year and was linked to St Mary's in the summer.
He scored again on Tuesday as Basel claimed a Champions League last-16 berth with a 2-0 win at Benfica - result which probably makes it tougher to tempt him away in the next transfer window.
Blick claim several other Premier League clubs are interested in the ex-Molde man with Leicester City, who were also linked in the summer, present to see him in action in Lisbon this week.
Elyounoussi, who is under contract until 2021, is expected to fetch several million euros when he does leave Basel.