Saints turned down a reported £5.5million bid from Derby for the Trinidad & Tobago forward, who has also been linked with a move to Sunderland.
Burley told the club's official website: "I want to keep Kenwyne. He came and spoke to me on Thursday, and I told him I wanted him to stay.
"With all this speculation, it has not been easy for him. Some managers go shooting their mouths off and talk in the press, but I don't think that is right.
"It will be totally my decision with Kenwyne.
"We don't need to sell him, the club is in a good financial position - which is why we can turn down £5.5million for a player."
Vi bruger cookies for at kunne give dig den bedste oplevelse. Ved at bruge vores side accepterer du brugen af cookies. Du kan læse mere i vores persondatapolitik.