Shapiro, who has been with the Indians since 1992, served as the club's general manager from 2001-10 before he was promoted to president.
Toronto has been searching for a replacement for current president Paul Beeston, who will be retiring at the end of this season.
The 70-year-old Beeston, from Welland, Ont., was the Blue Jays first employee in 1976.
The 47-year-old Shapiro has been a steady hand for Cleveland's organization, guiding it around financial obstacles while overseeing the renovation of Progressive Field.
The Blue Jays have major plans to overhaul Rogers Centre, and Shapiro could be intrigued by that challenge as well as joining a team that has overhauled its roster this season with trades and signings.
Shapiro is considered one of baseball's top executives and has served on MLB's competition committee.
Several media reports suggest that Shapiro's hiring could be announced as early as Monday.
Toronto hosts Cleveland for a three-game series starting Monday.
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