New Brighton signing Beram Kayal has paid tribute to Celtic fans for their backing during his time at Celtic Park.
The duo have been linked with the St Mary’s exit door following the departures of Rickie Lambert, Adam Lallana, Luke Shaw, Dejan Lovren and Calum Chambers.
But Krueger has dismissed speculation surrounding the pair's future after they were both linked with a reunion with former Southampton boss Mauricio Pochettino at Tottenham, insisting they remain central to the club's plans for the future.
"They are part of the core that we have decided to keep in Southampton," Krueger said. "Jay Rodriguez and Morgan Schneiderlin are not for sale.
"They will be a part of our club as we enter the new season, and in and around Jay Rodriguez and Morgan Schneiderlin and the rest of our core players, we’re excited to continue to build forward but all of those stories (linking the pair with a move away) are false."
When asked if he could categorically guarantee Southampton fans the pair would remain with the club, Krueger replied: "That's correct.
"It is come September that will be judgement day and I think that people need to realise we are in a process right now. The match is far from over and on September 1 let's have a conversation."
Rodriguez, who is still interesting Spurs and Liverpool, is understood not to have signed a five year contract tabled to him while Schneiderlin is understood to have met with the club on Tuesday to discuss his future.
Schneiderlin later tweeted: "6 years of an amazing journey #saintsfc DESTROYED in 1 hour !!!"
Sky sources understand Southampton will put in another bid for Celtic keeper Fraser Forster. Krueger insisted he sympathised with the club’s supporters over the raft of departures this summer but has assured them the club have sufficient money to rebuild the squad ahead of the new season.
"Now we have finances and once again we are excited to continue in the rebuilding process in the next few weeks," he said.
"We will have players coming in. We will make transfers to strengthen the squad. I can understand some frustration maybe in and around our fans but we needed to stay silent (on the departures) to bring in the best deals for the club.
"Ronald Koeman has done an outstanding job with his staff thus far in pre-season. We're extremely pleased with the work that’s going on at St Mary's. We know we are going in the right way but we also know we need more players and we need more depth in our line-up."
Sky sources understand Southampton hope to sign three players by the end of the week - a keeper, defender and forward, but Krueger is unwilling to put a time-frame on the completion of any deals.
"It's not about getting them in today or tomorrow, it's about getting in the right players on the right deals that fit the way Ronald wants to play and that we want to play," he said.
"Of course we would all like to see these players in sooner rather than later but we will make the deals at the time that we feel are right for the club.
"We are very close to a few deals and by the time we are close to kicking off against Liverpool of course for Ronald and his staff it's better that we get them in tomorrow so that he can begin the process of integrating them into the team and integrating them into the system of play.
"But we are working as quickly as possible to make that a fact. I don’t like to talk in definites when we don’t know them."