The 27-year-old is currently serving an eight-month ban from football after admitting 232 breaches of the Football Association's betting rules.
Toney cannot play again until Jan. 16 next year and was fined £50,000 ($60,900) but that has not deterred potential new clubs from launching a mid-season move.
Brentford are reluctant to lose Toney but the striker fuelled speculation over a transfer by revealing his admiration for Arsenal in an interview on the Diary of a CEO podcast last month.
"The next club I go to, if I was to move, would be the right club," he said. "I've been a Liverpool fan my whole life... But from young, I've liked Arsenal.
"I've liked watching Arsenal and how they play and how passionate their fans are. I'm a Liverpool boy at heart."
Sources have told ESPN that Toney recently changed agents with a view to the next chapter of his career.
Reports have suggested Brentford could demand around £60 million but once source close to the situation told ESPN that any fee involved is likely to be considerably higher, potentially in the region of £80m.
Negotiations have not yet taken place and Toney is concentrating on rebuilding his fitness after being allowed to return to Brentford training this month as per the terms of his punishment.
Arsenal are believed to be weighing up their attacking options and Toney, who scored 21 goals in 35 appearances for Brentford last season, is one forward they admire.
Chelsea are also keen on adding a striker having struggled for goals so far this season, scoring just five times in their opening six Premier League games.