On Tuesday, it was confirmed that Mauricio Pochettino had left to join Premier League rivals Tottenham on a five-year deal, the Argentine departing after less than 18 months at the helm.
Pochettino's move opens a vacancy at a club that finished eighth in the top flight last term, with young, English players forming the core of the side, and Krueger said on Wednesday that the process of finding a new boss was well under way.
"It's really important that we move swiftly on this," he told the club's official website.
"We've lost some time now since the season ended 16 days ago and I think it's in all of our wishes not to rush the situation but to have a certain speed to it so that we can move.
"The preparation that's been done by our scouts and recruiters make us feel we can move quite quickly to the next step to then present a shortlist of managers that we are considering, and then we can then move forward on the players, because we want the manager to make some decisions and be involved with who comes and goes, and they need to be able to continue to push the Saints forward."
Krueger went on to say that Southampton was a club attracting a lot of interest from would-be managers and players.
"We have the highest quality managers in Europe knocking on our door," he said.
"We also have the highest quality available players and their agents knocking on our door. There's a lot of exciting people that want to come here.
"Behind the scenes there are many people interested in what we're doing here. I believe the fact there's so much interest in our manager and our players is also a sign that we're doing the right things here in Southampton."
