The south-coast club are second in the Premier League after winning eight of their first 12 games and suffering only two defeats in a sensational start under Koeman.
Southampton had been tipped for a season of struggle after losing a host of key players, who followed manager Mauricio Pochettino through the St Mary's Stadium exit door.
But rather than bracing himself for a relegation battle, Koeman has set his sights on securing a UEFA Champions League spot in his first season in English football.
The Dutchman knows the visit of champions Manchester City will be a major test for Southampton, but the former Netherlands defender is backing his side to rise to the challenge.
"We realise that this Sunday we play against one of the teams who can win the title," said Koeman.
"We've shown before that it's not easy to beat Southampton, and we'll show that on Sunday.
"If we want to do something special this season, then we have to win these types of games.
"They have a strong squad and good players. In the last few weeks [Sergio] Aguero has made the difference.
"Manchester City will take this game seriously. They know that we are strong and that they'll have to be good to beat us."
City will go above Southampton into second spot if they consign Koeman's men to a first home defeat of the season.
Manuel Pellegrini's side head into the game on the back of a dramatic 3-2 UEFA Champions League win over Bayern Munich in midweek, which kept their hopes of qualifying for the last 16 alive.
Southampton's impressive start to the season has led to them being described as a surprise package, but City boss Pellegrini's eyebrows have not been raised by their impressive form.
The Chilean said: "Southampton are a good team with very good players and they're playing very well. It's no surprise they are where they are.
"Koeman has lost four or five players but they are still a good team and that's a merit to him. It will be a tough game."
Southampton have conceded only one goal at St Mary's this season - against QPR in September - but they will have to be at their best to keep in-form striker Aguero quiet.
The Argentina forward scored a hat-trick against European champions Bayern and has 18 goals to his name already this season.
City will be without Edin Dzeko (thigh), David Silva (knee) and Aleksandar Kolarov (calf), but Yaya Toure and Fernandinho are back in the squad after missing the win over Bayern due to suspension.
Southampton midfielder Steven Davis is rated as doubtful due to a hamstring problem.
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