The talented Brazilian was a long-term target for AC Milan over the summer but a deal failed to materialise.
In recent weeks Chelsea, despite the transfer ban placed upon them, have been mooted as serious suitors of his signature.
Jimenez is confident Fabiano's heart lies with the Andalusian club but admits the lure of West London does concern him.
"I worry about the interest of Chelsea but I don't worry about what Luis Fabiano thinks," said Jimenez.
"He is a great guy and I have no doubt that he thinks only of Sevilla.
"The problem is for Chelsea and all the other teams in Europe because Luis Fabiano plays for Sevilla.
"Our desire is that he stays in Sevilla for his whole career."
Sevilla president Jose Maria Del Nido has conceded the player has a release clause in his contract, thought to be 30million euros, but he is nonetheless determined to keep him.
He said: "I'd like to keep hold of Luis Fabiano until the end of the season and hope he scores 40 goals with us."