Wilson has taken up the role of sporting director at Ibrox, having previously been at Watford, Huddersfield and Falkirk before joining the Saints.
The 36-year-old played a key part in bringing Hasenhuttl to St Mary’s Stadium in December 2018 following the departure of Mark Hughes.
Austrian Hasenhuttl, though, believes the Southampton board will continue the work behind the scenes as he looks to keep the club in the Premier League.
“It was surprising for me, but what I can say is that I had a very good relationship with him,” Hasenhuttl said.
“I liked to work with him, he is a nice and fantastic character, a fantastic personality and I enjoyed every minute with him.
“I wish him all the best. He goes to a famous club to a big club and has a new challenge. It was, I think, about five years here.
“For us, it is a big loss for sure – but this is what happens in football. For us it will be important that we bring somebody new in and we will do this.
“This will be the main work for the board and for (chief executive) Martin (Semmens). They are still busy in this part I am sure.
“I have other things to look at and work to do, but I will also have my opinion for this position.”
Hasenhuttl said at a press conference: “We have our plan, our schedule, what we want to do, the players we want to look at.
“It is independent from Ross. We know what we want to have for our game, for the positions we need and so that doesn’t make a big change in a moment.”
Southampton will resume domestic action at Wolves on Saturday looking to produce a response following the 4-1 home defeat by Chelsea, which was a third straight Premier League defeat.
Hasenhuttl reported no fresh injury problems after international duty, however, full-back Cedric Soares and winger Moussa Djenepo remain sidelined.
Southampton continue to monitor Djenepo’s progress from a niggling hip muscle problem but will not rush the Mali forward – a summer signing from Standard Liege – back into action.
“Moussa can run, he can sprint, he can pass the ball – but shooting, so far we didn’t do with him because this is the most dangerous movement he can do,” the Southampton boss said.
“We try to keep it away because when sometimes it is hurting again then it can take longer that he is out.
“I think we must pay attention, we must be very, very patient with him. Today he has a scan again, and then know more how long it will take.
“At the moment, I am thinking about week of the (two) Manchester City games, that is maybe the hope I have.”