The striker could be sold in January once he has completed an eight-month ban for betting breaches and is attracting interest from both the Gunners and the Blues.
The Bees are ready to demand as much as £80m for the 27-year-old, who recently signed up with super-agent Jonathan Barnett ahead of the January transfer window.
Toney returned to training last week but cannot play until January 16 after receiving his ban in May.
He is believed to be open to joining a bigger club in an effort to force his way back into the England reckoning ahead of Euro 2024.
Despite spending over £1bn since in charge of the Blues, Chelsea co-owner Todd Boehly is prepared to return to the market in an effort to solve the club’s goalscoring woes.
Arsenal, meanwhile, are weighing up a move for a new forward to compete with Gabriel Jesus and Eddie Nketiah as they attempt to challenge Manchester City for the Premier League title.
Speaking ahead of Brentford’s Carabao Cup clash with the Gunners, Frank said: “For me as a coach, I don’t want to sell him.
“Of course I want my best players. Of course I say to Phil [Giles] or Matt [Benham] ‘don’t sell him’.
“Ivan is a world class player, so why should we sell a world class player? I want him to play for Brentford for the rest of his contract but that’s not up to me to decide.”