The 26-year-old was signed from Celtic to take over the number one position at St Mary's and impressed on his home debut in Saturday's goalless draw with West Brom.
The former Newcastle man is hoping to displace Manchester City's Joe Hart in Roy Hodgson's England team.
Fraser Forster: "It was good to keep a clean sheet.
"It was nice to make the save, it is one of those where I haven't really had that much to do save-wise and it is just important to try and help the lads out as they defended brilliantly in front of me.
"It is my job when one comes along to make those saves.
"I was delighted to have the interest from Southampton. They were brilliant last season, a few have gone out but we have got a lot of good players in.
"It is one of them where I have only played two games in the Premier League and it is a fantastic league to be a part of.
"It comes down to playing well for Southampton first and foremost - that is what it is all about. If I do all that then we will just have to see what happens with England.
"For me personally it is about me working as hard as I can with Dave (Watson, Southampton and England goalkeeping coach) and in the gym and improve myself as a keeper."