17-year-old Shaw is reported to be a target of Arsenal and Chelsea in the winter transfer window after impressing in his seven appearances for Southampton so far this season.
The defender's inclusion has coincided with a period of only conceding six goals - in vast contrast to their early season defensive troubles where they shipped 26 goals in their previous nine matches.
The young left-back has already been likened to Tottenham star Gareth Bale, who also came through Southampton's academy, by team-mate Adam Lallana.
Cork, himself a Chelsea academy product, said: "I think the best thing for Luke would be to stay here - everything is set up for him to play in the Premier League.
"He will only improve here and playing will help him. As long as he works hard, he will go as far as he wants."
Saints boss Nigel Adkins had already stated his intention to hang onto Shaw earlier this month, saying: "Luke is very happy where he is. He is in the first team, he has got great people around him and his future is very bright.
"If Southampton are in the Premier League, there is no better place for any of our young players. They have more chances of first team football here than elsewhere."
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