The 24-year-old Frenchman was reportedly the subject of heavy interest from Tottenham and was said to have handed in a formal transfer request to try to force through a move.
Schneiderlin publicly expressed his frustration on social media when the request was rejected but has settled well this season, impressing as the Saints sit second in the Premier League.
And Southampton boss Koeman has stated that Spurs did not make a serious bid for the midfielder, who is now happy with the ambition shown by the club in the transfer window.
"Tottenham never made a serious bid," Koeman told reporters. "A lot of players left the club and Schneiderlin maybe had some disappointing moments but I spoke to him.
"We wanted to do everything to show our ambition and we showed him that with the signings we made.
"Maybe he had some doubts about the club’s ambition but if you ask Morgan now about our ambition, it is what he would like to see.
"He is a key player. He scores, he defends, he attacks and with that kind of player you like to do everything to keep him in the team."
The Saints travel face Tottenham at White Hart Lane on Sunday.