Multiple offers are expected for Toney when he has completed his eight-month ban for breaching betting rules.
Brentford are said to be resigned to losing Toney next year but are expected to demand £80m for the 27-year-old.
Tottenham and Arsenal have both been linked with Toney, who scored 20 Premier League goals last season.
“It is very timely for me to be joining CAA Stellar”, Toney said. “It feels like a fresh beginning. Never underestimate the power you have to take your life in a new direction.”
Barnett said: “I am very excited that we have signed Ivan. He is a great talent and I am sure this is the beginning of a fantastic future for him.”
Toney joined Brentford for an initial £5m in 2020 and his contract expires in 2025, so he will be in the final 18 months of his deal when the January transfer window opens.
He is banned until January 16 after admitting breaches of the FA betting rules.