"We will sell nobody in January. Nobody. Even when they pay 100 million (pounds)," Koeman said at his pre-match news conference on Friday.
The Dutchman added that despite the reported interest, Mane was not yet the finished product.
"I think he (Mane) can still improve in some cleverness to know what's his position on the pitch," Koeman said.
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"I think still he can be more dangerous if he knows exactly when to have that movement in the space behind the defenders.
"And in front of the goal he can be a more scoring player, but he's young, a great player, and it's a pleasure to work with (him)."
Tenth-placed Southampton host bottom side Aston Villa on Saturday.
They won the corresponding fixture 6-1 last season when Mane grabbed the fastest hat-trick in Premier League history, but Koeman is expecting more of a fight from Remi Garde's men.
"They are in a difficult, dangerous situation in the table. They need the points and they will react because they have qualities," he said.
Saints captain Jose Fonte has returned to training after recovering from the knee injury he picked up against Manchester City at the weekend.
He could return to the heart of Southampton's defence against Villa, after missing the 6-1 drubbing by Liverpool in the Capital One (League) Cup in midweek.
(Reporting by Simon Jennings in Bengaluru; editing by Toby Davis)
