After Argentine Mauricio Pochettino quit the Saints on Tuesday to join top-flight rivals Tottenham Hotspur, Krueger says the club is poised to act "very swiftly" in appointing a successor.
"Les Reed (the club's football director) is inundated at the moment with requests from high quality managers all over Europe trying to come to Southampton," he told Sky Sports News.
"We are building a long list which should quickly become a short list. Any club that is well-constructed and well-organised is permanently tracking potential managers for a situation such as this."
Pochettino was put in place by Krueger's predecessor Nicola Cortese, but Krueger said the same team which had found and appointed Pochettino were tasked with selecting his replacement.
Former FC Basel boss Murat Yakin is the bookmakers' early favourite.
Krueger would not speculate, but did say the club was open to another non-English manager.
"Being Swiss or German or from England is not relevant, we do not have an official list, but are going to move very swiftly - it does not matter what language, or what ethic background - we are looking for a manager to fit Southampton FC now.
"We are optimistic and we are certain that the foundation remains intact here. Southampton is in a strong position and we will continue to move forward."
Saints finished the season in eighth spot, equalling their best finish with a record points haul.
While Pochettino' exit has been expected in some quarters to herald an exodus of its leading players, Krueger said there are no offers on the table for England World Cup players Adam Lallana or Luke Shaw.
Saints captain Lallana is reported to be a 20 million pounds-plus target for Liverpool, while Manchester United are said to be tracking left-back Shaw, with a price tag of 25 million pounds reported in local media.
"Under the circumstances we wanted to wait to clarify the manager's situation to move forward. There are no deals done," Krueger said.
Pochettino has been linked with moves to bring Saints striker Jay Rodriguez and defender Nathanial Clyne to Spurs, and a swift Southampton appointment would help settle the squad and repel circling suitors.
(Writing by Ossian Shine; Editing by Amlan Chakraborty)
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