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Ronald Koeman believes Southampton can qualify for the Champions League.
Koeman's side broke into the top six of the Barclays Premier League after Shane Long struck in the second half to extend their excellent run with a 1-0 victory at Swansea.
The south-coast club have won five and drawn one of their last six games to move within a point of fifth-placed Manchester United and seven of Manchester City in fourth.
And Saints boss Koeman admits their ambitions have broadened in the last few weeks, with a top-four spot to qualify for the Champions League in their sights.
"If Leicester can win the title, then we can finish fourth or fifth," the Dutchman said.
"We are only one point below Manchester United after today and my team is looking very comfortable.
"We had a difficult start to the season and a difficult period in December, but we are back fighting and looking to finish as high as possible in the table."
Southampton's rise has been based on an impregnable defence which has not conceded a goal in six games.
That run has coincided with the return of Fraser Forster after the England goalkeeper missed 10 months with a serious knee injury.
"We didn't play so well in the first half but we played better football in the second half," Koeman said.
"We scored a great goal and we showed a great attitude.
"It was a good cross and good positioning in the penalty area for the goal.
"Shane had a free header but he had still had to score."
Swansea remain three points above the relegation zone following their first defeat under Francesco Guidolin.
Long's ninth goal of the season ended Swansea's four-game unbeaten run, and the Italian head coach expressed his disappointment over the performance.
"For the first time in my five matches here I saw my team play not very well," Guidolin said.
"I don't know why we did not play well. This is my first month and it is difficult to understand anything.
"I think we can do better but Southampton are a good team and are difficult in some situations.
"They are playing well in this period with five victories and six clean sheets and score against a lot of teams, not just us."
Swansea created little to trouble Forster and company, but they had a strong penalty shout waved away in the final minute.
Southampton substitute Maya Yoshida appeared to bundle over Alberto Paloschi in the box but referee Jonathan Moss was unmoved.
But Guidolin said: "It could be (a penalty) but I don't speak about it.
"My culture is not to speak about the referee because his job is difficult, like my job and that of the players. That's my culture."