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.”