Morrow suffered the injury during the first inning of Monday night's game against Washington. He was removed after tossing just nine pitches, and manager John Farrell later described the injury as "probably significant."
There is no immediate timetable available for Morrow's return but he will be out for at least the rest of the month.
"The further tests and the imaging indicated that the oblique is strained," Blue Jays manager John Farrell said. "It eliminated any fractures in the bottom rib. So at this point, there's really no change as far as what our projection is. Which, I don't even know that we have an end date in mind right now.
"It's probably too early to determine. As sore as he showed up to the ballpark today, there's some significant discomfort."
Losing Morrow for an extended period of time would be a massive blow to the Blue Jays' aspirations of reaching the postseason. The 27-year-old has established himself as the club's best pitcher while also ranking among the best the American League has to offer.
Morrow is 7-4 with a 3.01 ERA in 77 2/3 innings this season. He also has struck out 67 batters while walking just 24 to go along with a Major League-leading three shutouts.
"He's pitched great, but it happens," Blue Jays general manager Alex Anthopoulos said. "And we're certainly not the first team to go through it. A lot of other teams do. It's a long year, and you just have to fight through it. And hopefully we can get him back soon, and we can make up some ground while he's away."
With an off-day coming on Thursday, the Blue Jays won't need a fifth starter until either June 18 or 19. The two main candidates appear to be right-hander Jesse Chavez and left-hander Brett Cecil.
Chavez, who was claimed during the offseason, was set to start for Triple-A Las Vegas on Wednesday night. He is 6-2 with a 3.68 ERA in 11 starts this season and appears to be the early favorite.
"He's definitely the frontrunner, I guess I would say that," Anthopoulos said. "But we haven't made a commitment one way or the other. He's scheduled to pitch [Wednesday], but we don't have to make a decision for awhile. But there's no question that he has probably been the best starter down there, but I certainly don't want to lock into that name. But he's put himself in a good position."
Morrow's departure will create a void in the starting rotation, but Anthopoulos said the injury won't put any pressure on the organization to make a trade.
"You're hopeful that whatever we end up doing, whatever time he does miss, you might get someone that gets hot and does well," Anthopoulos said of his internal options. "I know when Carlos Villanueva had to step into the rotation last year ... for probably three or four weeks, he did a very good job, and then he got hurt and didn't do as well.
"But you never know. Someone steps in, they might do a good job for you, we might score some runs for them and end up getting some wins. It's one start every five days. There's value to those starts because of Brandon's ability to work deep into games, save the bullpen, things like that. But at the same time, it is 25 guys and there are lots of teams in this division and in this league that have lost a lot of good players and have been able to overcome it. Hopefully, we'll be in the same boat."