Shapiro's hiring will be announced early this week, per Heyman, who notes that it could come as soon as Monday.
Shapiro is currently serving as the Cleveland Indians' president of baseball operations and is widely regarded as one of the top baseball executives in the game.
He has been in the Indians organization since 1991 and served as their GM from November 1, 2001 to October 3, 2010, when he was promoted to president.
During that time as GM, Shapiro won the Sporting News Executive of the Year award twice, in 2005 and 2007, leading the Indians to the American League Central Division crown the latter year.
According to Heyman, the Shapiro and the Blue Jays plan on keeping Alex Anthopoulos as general manager. He has built the team towards their current run, where they are leading the American League East and pose as threats to win the World Series.
Shapiro's exact starting date with the Blue Jays is not known. He will be replacing Paul Beeston, who is planning on retiring at the end of the year.