The Norwegian international is thought to be a direct replacement for Dusan Tadic, who left Saints for Ajax.
Elyounoussi scored 24 goals during his two-year stint at Basel and Saints have apparently made no secret of admiring the winger during that time and his earlier career forged in Norway.
“It’s good because they’ve been keeping an eye on me since 2012,” he said.
“It’s been a long time and that makes me confident that I can trust the club and come here and be myself because the only thing they are expecting from me is to do the same in Basel; to play my game and to try to create chances.
“It’s almost a little bit creepy that they’ve been stalking me for so long, but that means a lot that they’ve been tracking me like that.
“They know my qualities and now I’m here to get better as a footballer. I’m really sure that I’m going to do it here.
“They know me and the system suits me as well, so for players coming in it’s something to think about.
“For me it was a really easy decision to come here. It’s the best league in the world. The progression has been good so far. The boys have been treating me really well and I’m really happy to be here.”
Elyounoussi also said that having some fellow Scandinavians at Southampton has helped him easily settle at the club.
“It’s easy because a lot of the players are the same age as me and there are now a couple of Danish players so I can speak Norwegian with them.
“It’s a really good group. That’s what I’d heard before I came here and I can only say that that was true.
“The people around the club as well are treating me really well and I’m looking forward to being part of the club and I’m excited to get started.”