Bluebirds manager Malky Mackay confirmed that the 31-year-old is considering a loan offer from the Canadian club, and gave his blessing to the deal.
"He's going over to have a look around and a talk with Toronto about the possibility of a loan," Mackay told the South Wales Echo.
"Whether it happens or not is another matter, but we've given him the opportunity to do that. Rob needs his career to go on because this is football after all.
"There's nothing been biting here for him and if he feels that a move there would be something beneficial to him, then I'm certainly not going to stand in his way. I'd never stand in a player's way when it comes to them getting the chance of playing football."
Earnshaw, who went out on loan to Israeli side Maccabi Tel Aviv last year, has not made a single appearance for the Championship outfit this season.