The broadcaster reported the trade first, followed by ESPN, The New York Times, the Sporting News and others.
Citing MLB sources, the outlets said the nine-player deal would send Dickey, catcher Josh Thole and a minor-league prospect to Toronto.
In exchange, catchers Travis d'Arnaud and John Buck, pitching prospect Noah Syndergaard and another unnamed prospect would reportedly move to the Mets.
The trade is dependent on whether the Blue Jays can negotiate a long-term deal for the knuckle-balling Dickey, who went 20-6 with a 2.73 ERA in 2012, pacing the National League in innings, strikeouts, complete games and shutouts.
The deadline to reach a contract deal is Tuesday, Fox Sports said.
The prize for the Mets would be 23-year-old D'Arnaud, considered among the best hitting prospects in the minor leagues. He batted .333 with 16 homers in 67 games at Class AAA Las Vegas before ending his season in June with a torn left knee ligament.
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'
Kate Middleton recycles dress at movie premiere