Saints have qualified for the group stages of the competition, meaning at least six additional matches from mid-September until early December.
"We've got players who've played in the Champions League," said 28-year-old Forster.
"We're in great shape. We want to have a real go at it next season."
Forster, who signed a new contract this month, is currently training with England before Euro 2016 starting on 10 June.
The former Celtic keeper pointed to players such as central defender Virgil van Dyke as being important to Southampton's European ambitions.
"Maybe a couple of players will come in and a couple will go in the summer," said Forster.
"I don't think we need a lot to add to the quality that we've got."
Meanwhile, Southampton have confirmed the departure of keeper Kelvin Davis, midfielder Gaston Ramirez and Will Brit, another keeper, after their contracts expired.
Davis, 39, has retired following 10 years at the club, while Uruguay midfielder Ramirez, 25, leaves after a four-year stay, which included a loan spell at promoted Championship club Middlesbrough this season.
Goalkeeper Fraser Forster says Southampton have enough experience to cope with the demands of the Europa League next season.