Southampton come up against former head coach Mauricio Pochettino and his Spurs team at White Hart Lane this weekend, having made a flying start to their campaign.
Despite the loss of Pochettino and key players such as Adam Lallana, Rickie Lambert, Dejan Lovren and Luke Shaw, Southampton sit second in the league.
With just one top-flight defeat to their name this season, Southampton are five points ahead of Spurs and Koeman feels his side could finish above Pochettino's men.
"If we continue like this, there is a possibility but normally teams like Tottenham, Manchester United will be up [in the table] because they have enough very good players," he said on Friday.
"That makes the club big, the name is big, but we try to do it our way. If we keep the belief and keep the spirit nothing is impossible.
"It was a difficult start to the season with all the changes in the team. We're enjoying very much our start.
"It's a long season but we know our qualities. The most important thing is to keep the spirit and our way of playing. What we showed [in the 2-1 defeat] against Liverpool is the example of how we have to play.
"We'll try that this Sunday and see what happens."
Koeman also sought to play down the occasion of the club coming up against their former boss, although he did concede it would be a special occasion for some of his players who played under the Argentinian.
The 51-year-old is eager to try and top Pochettino's achievements of last term and finish higher than eighth in the Premier League.
Asked if he felt it would be a special day, he said: "For me, not. I wasn't here last season but I understand for Mauricio Pochettino it may be special and maybe for some players.
"It's always special for some players who played with Mauricio last season but there is no influence in the final result.
"From the first day of this season, what I saw was a great spirit in the team and I heard that was one of the qualities in the last season.
"The challenge now is higher than the last season. Everyone is surprised by eighth last season but people love how we're doing and we want to keep doing that.
"We will try [to top eighth] and see if we can do it but there's a long way to go."
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