Schneiderlin had said he wanted to follow other leading Southampton players like Adam Lallana, Rickie Lambert and Luke Shaw out of the South Coast club ahead of the Premier League season which kicks off this weekend.
He had been widely tipped in media reports to join his former Southampton manager Mauricio Pochettino at Tottenham Hotspur and told Koeman last week he was "not mentally or physically ready" for a friendly against Bayer Leverkusen.
But the Dutchman, who was appointed as Pochettino's successor in June, told the BBC: "He showed me he is ready to play in the last week of training. I expect him to stay because he is not for sale."
Despite a traumatic pre-season in which so many players have left for bigger clubs like Liverpool, Manchester United and Arsenal, Koeman said he was looking forward to Sunday's opening game at Anfield.
"It's a great start for us to play away against Liverpool," he told a news conference.
"We know it's a tough one, but everybody starts on nil. Everybody has to show their qualities and ambitions and I believe we are very well prepared."
- Sports & Recreation
- Mauricio Pochettino
- Rickie Lambert
- Adam Lallana
- Tottenham Hotspur