The Guardians selected Norris' contract from Triple-A Columbus on Monday.
He'll be joining the Guardians as a replacement on the 40-man and 26-man active rosters for right-hander Noah Syndergaard, who was designated for assignment following Sunday's 10-7 win over the Blue Jays. The 30-year-old lefty reliever is set to begin his third stint of the season with the Guardians after pitching to a 3.38 ERA, 1.59 WHIP and 10:11 K:BB in 10.2 innings between his previous two terms.