"Looking forward to a new chapter with the Jays," Dickey wrote on his Twitter account.
In exchange, catchers Travis d'Arnaud and John Buck, pitching prospect Noah Syndergaard and another unnamed prospect would reportedly move to the Mets.
Fox Sports said the trade was dependent on whether the Blue Jays could negotiate an extension 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.
Several reports said Monday Dickey agreed to a two-year, $25 million deal with the Blue Jays, pending a physical.
The prize for the Mets would be 23-year-old D'Arnaud, considered among the best hitting prospects in the minor leagues.