The south coast side have spent the last seven years outside England’s top-flight but have returned with lofty aspirations following back-to-back promotions
"What we've got to do is make sure we're ambitious in the Premier League and not just go there and think 'oh, well if we can just get enough points to stay up then that will be successful',” he told reporters.
The south coast outfit have been out of the Premier League since its 2005 relegation, and even fell as far as the third tier.
However, Adkins led the club to second place in League One before repeating the feat in the Championship last season and the former Scunthorpe manager is now bullish on his side’s chances.
"We've got to drive on, strive to be the best and Europe is obviously the goal," Adkins said.
"Southampton Football Club ultimately have got to be in the division to get into Europe and ultimately have the dream of winning the division."
All three promoted sides stayed in the Premier League last season and Adkins believes they have set a benchmark for his club, Championship winners Reading and playoff victors West Ham.
"We can draw inspiration from Norwich and Swansea last season in how they acquitted themselves,” he continued.
"Nicola Cortese [chief-executive] set the dream out and the vision to me to be in the Premier League within five years.
"Since he has been here, they've done it in three with back-to-back promotions and we're very, very ambitious.
"It's a very positive future ahead.”