Ronald Koeman's side have exceeded expectations in the Premier League and currently sit third in the table, two points clear of Manchester United in fourth.
Sunday's 1-0 win at United continued a run of good form for Southampton that has seen the club pick up 13 points from a possible 15.
Following victories over the likes of Everton, Arsenal and United, and a 1-1 draw with leaders Chelsea, morale at St Mary's is understandably high and Davis feels a top-four finish is achievable.
"We're growing as a team and we know we can come against these teams and compete on a consistent basis," Davis told Southampton's official website.
"I think if, when we get to the end of January [to] end of February, and we're in a similar position then we can start to realistically go at it [the Champions League].
"We're in a great position, we're enjoying our football, every game we look forward to and we can't ask for anything more at this point in time.
"It's just about trying to keep that consistency and keep the desire within the team to go out and get results."
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