McKenzie (shoulder) threw a bullpen session that went well Thursday, and he's expected to throw one more before stepping away from his rehab to attend his brother's graduation Tuesday.
McKenzie will then return to Cleveland to be evaluated and scheduled for a rehab assignment. The 25-year-old is recovering from a right teres major strain, and he'll likely need multiple rehab starts before he is cleared to return to the major-league roster. He's eligible to return from the 60-day injured list May 29, but it is possible his rehab may extend beyond that date depending on how he performs and when the assignment starts.