The Saints are currently the Premier League's most in-form team, taking 16 points from the last 18 available and not conceding a single goal in that time.
A 1-0 victory over Swansea City at the Liberty Stadium this afternoon saw Southampton close to within one point of fifth and seven of fourth, and Koeman is taking inspiration from unlikely title challengers Leicester City in his quest for lofty ambitions of their own.
"If Leicester can win the title, then we can finish fourth or fifth. We are only one point below Manchester United after today and my team is looking very comfortable," Koeman told reporters.
"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. We didn't play so well in the first half [of the win over Swansea] but we played better football in the second half.
"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."
Southampton's next league game sees them take on Chelsea at St Mary's on Saturday 27.