Schneiderlin, 25, was linked with a move away from the Premier League club throughout the close-season.
Having helped Southampton into second in the table through 11 games, Schneiderlin continues to be linked with a transfer.
Speaking to a news conference on Tuesday, he reiterated his desire to play for a "big club".
"A career passes quickly, my aim is to play for a big club," Schneiderlin said while on international duty with France.
"Today I'm at Southampton, we are second of the league, all is going well. We will see at the end of the year what will happen, where we will be with Southampton, if we will be in the same place and qualify for the Champions League.
"And I will make choices at the end of the year with my club."
Schneiderlin's team have conceded just five league goals this season to be four points behind leaders Chelsea.
It came despite several close-season departures, and Schneiderlin admitted even he was surprised.
"If at the end of transfer market, you would have told me that Southampton would be second in the Premier League, I would not have believed you," he said.
"So I guess for you, outside [of the club], it's even more surprising.
"But even if we lost a lot of people this summer, including the coach, the club has recruited well. There is a new coach who came with a new philosophy and who is getting used well to the club.
"We still play good football. The new players have a good state of mind, and have been immediately effective. Their integration has been rapid.
"We have a strong team, we concede just a few goals, and we play good football. We get along very well. This is our strength."
