The right-hander, who has been sidelined since mid-June with a left oblique strain, will throw live batting practice on Wednesday, and, if all checks out, will then be ready to take the mound for the first time since his injury.
"Providing all goes accordingly, [he] would begin a rehab assignment on Sunday," manager John Farrell said.
Morrow was forced to leave his start against the Nationals on June 11 following just nine pitches after feeling something in his side, which ended up prompting a visit to the disabled list.
The 27-year-old was Toronto's best pitcher at the time of the injury and was well on his way to a career season.
Over 13 starts, Morrow went 7-4 with a 3.01 ERA, 1.00 WHIP and three shutouts, after entering the season having only achieved that feat once before.
Farrell suspected that Morrow would need to make three-to-five rehab starts before the team would activate him, so a mid-August return would likely be a best-case scenario for the club.