According to media reports Schneiderlin and Rodriguez have been close to joining Tottenham Hotspur, adding to a player exodus that has left Saints fans concerned about their ability to compete this season.
Chairman Ralph Krueger has already sanctioned the sales of Rickie Lambert, Adam Lallana and Dejan Lovren to Liverpool, Calum Chambers to Arsenal and Luke Shaw to Manchester United.
Those deals have dismantled the squad that finished eighth last season under Mauricio Pochettino who left St Mary’s in the close season to take over at Tottenham.
But Krueger gave a guarantee to Southampton fans that French international Schneiderlin, 24, and England’s Rodriguez, 25, would be staying on the south coast.
“That’s correct,” he told Sky Sports News. “They are part of the core that we have decided to keep in Southampton – Jay Rodriguez and Morgan Schneiderlin are not for sale.
“All of those stories are false,” added Krueger of the Spurs reports.
“It is come September that will be judgment day,” he said in reference to the end of the transfer window. “I think people need to realise we are in a process right now.”
Krueger added that manager Ronald Koeman had money to spend and that new players would come in during the window.
“Now we have finances and once again we are excited to continue the rebuilding process in the next few weeks,” he said.
“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 outgoings) 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 the right way but we also know we need more players and we need more depth in our line-up.”
(Writing by Stephen Wood, editing by Tony Jimenez)