Chairman Ralph Krueger says Southampton are already attracting "huge interest" from managers looking to replace Mauricio Pochettino.
Pochettino was named Tottenham's new boss on Tuesday and Krueger is hopeful the club can replace him quickly.
"We have the highest-quality managers in Europe knocking on our door," Krueger told BBC Radio Solent.
"It's important we move swiftly. It's in all our wishes not to rush the situation but to have a certain speed."
Krueger continued: "We feel we can move quite quickly to the next step and have a shortlist presented to the public of which managers we are seriously considering."
Les Reed, executive director at St Mary's, is leading the search for a new manager - a process they hope to conclude before this summer's World Cup.
Pochettino could be followed out of St Mary's by several key players, with Liverpool pursuing Adam Lallana and Dejan Lovren, while Manchester United have made a £27m bid for Luke Shaw.
But Krueger says any player exits will be on the club's terms, with all three tied to long contracts.
"We are in control of the situation, that is important for our fanbase to hear," he added.
"We will not end up with a summer without transactions, but I want to underline they will be on our terms.
"We have the highest available players and their agents knocking on our doors."
Krueger also rejected claims that the club could have done more to keep Argentine Pochettino, saying they did "everything possible" to ensure the 42-year-old stayed at Southampton, including offering him a new deal.
"We did everything in our power," he added. "There was a strong contract on the table and I don't believe that was the issue.
"I don't want to speculate on the reasons for the final decision. We respect and accept it."
The former NHL ice hockey coach also believes fans should be excited about the work being done on and off the pitch, which includes the arrival of a new chief commercial director, a new financial officer and a new marketing head.
He added: "We have 20 people working on scouting and recruitment all over Europe. These people are not out there playing cards. They are working, analysing video, collecting data and are not just doing that for players.
"This same group was involved in finding Mauricio Pochettino. They are out there recruiting, searching and building profiles and they have continued to do that because that is their job.
"I don't believe a club recruits 20 people to work in scouting if they don't want to improve and play European football."
Meanwhile, club legend Matthew Le Tissier has laughed off suggestions the club will go into meltdown following Pochettino's exit, amid rumours of player departures.
"We keep producing players and if you lose a couple, we've got players that can come through and take their places," he said.
"If we do lose a couple, we'll have a lot of money there to go out and spend, so I think, for me, if you appoint a decent manager who is going to use the money wisely, then I don't see any reason why we can't have a similar season to last season."