Swiss newspaper Blick claimed Saints sent a scout to watch the 23-year-old Norwegian in action for Basel earlier this month with the spy reportedly filling several pages worth of notes on the player.
Elyounoussi has impressed this season with five goals and nine assists for the Swiss champions, who reached the last 16 of the Champions League and now face Manchester City in 2018.
Leicester City and Newcastle have also been linked to the ex-Molde star and Elyounoussi is starting to believe he will soon realize his dream of playing in one of the top leagues in the world.
He told VG: "I can honestly say I know about interest from different leagues, including the Premier League. There are no rumors anymore. I know about interest from other places as well. It's a good position to be in. I am happy with that."