Ogunjobi has been dominant up front against the run for the Browns (3-1), who lead the NFL with 11 takeaways. A third-round pick in 2017 from Charlotte, Ogunjobi got hurt in last week's win over the Dallas Cowboys and didn't practice this week.
Rookie Jordan Elliott, a third-round pick from Missouri, will take Ogunjobi's spot.
"We are counting on him," coach Kevin Stefanski said. "He is pretty responsible. He follows through on study, and he makes sure he is in the right spot and right place. We just need him to go out there, and then whatever is asked of him, he has to do his job.
"Everything we have thrown at him, he has been ready to roll. For a young player, I think he is ascending."
The Browns could also be without starting safety Karl Joseph, who hurt his hamstring in practice Thursday and is questionable.
"We just have to use the next 50 hours or so to see how he responds and take it from there," Stefanski said. "I do not know exactly how it happened but some of these muscle injuries, that is just what happens."
If Joseph can't play, Ronnie Harrison, who came to Cleveland in a trade last month from Jacksonville, will start against the Colts (3-1).
"He is ready," Stefanski said. "He has been here now over a month. He studies hard. He is ready if called upon."
Cornerback Greedy Williams, who was projected as a starter this season, will miss his fifth straight game with a shoulder injury suffered in training camp. He has been practicing the past few weeks, but his shoulder isn't ready for game contact.