The 28-year-old has been showing a number of good displays ever since returning from a serious knee injury that kept him sidelined for an extended period.
He signed a four-year deal when he arrived from Celtic back in 2014, but the Saints are ready to ensure his stay even longer in order to avoid bigger clubs poaching the goalkeeper in the summer.
Koeman admitted that keeping hold of Forster in the summer is a big possibility, while also hinting at new deals for Victor Wanyama and Sadio Mane.
“That’s a possibility, yes,” he said.
“We will try to hold onto everyone of course.
“But we know our situation and we know we can’t win against the big teams with a lot of money.”
Wanyama and Mane has been linked with a move to Arsenal and United respectively, with the club hoping to avoid a repeat of their transfer raid back in 2014.
The goalkeeper has so far featured in 12 league matches this term due to his injury, but has already kept six clean sheets since his return.