The Netherlands international joined the club from Celtic earlier this season and has established himself as an integral part of the side.
Koeman has been impressed with the performances of the 24-year-old, but believes he can now go on to become a more complete player.
“He is doing well and in general I am very pleased with Virgil, but I think he can be better, he can be stronger, he can be still more complete as a defender,” Koeman told reporters ahead of Saturday's trip to Stoke City.
“He’s important, but he can be more important. More leadership, still more focused every day.
“I think he can still improve into making the next step as a player and that means every session maximum. Maybe his one weakness is that sometimes he looks a little bit sloppy, and that’s a mentality, and I like to change that and we have talks about that.”
Koeman agreed that there is a comparison to be made between van Dijk and Toby Alderweireld, who enjoyed an extremely successful period on loan at Saints from Atletico Madrid last season.
“I think you can compare both players,” said Koeman. "Alderweireld had a very good season in Southampton, he got the possibility in Southampton to show his qualities, but we like to keep Virgil for a longer time than with Alderweireld."