This is the second time Brazill has been suspended. He was banned for the first four games of the 2013 season for violating the same policy.
Brazill told reporters at training camp in July 2013 after it was revealed that he tested positive for drugs that "it's either money or marijuana. I know any of you would choose money any day." He failed to choose money again.
A Colts spokesman said the team does not have a comment on Brazill's suspension.
Brazill was expected to go into training later this month competing with Da'Rick Rogers and Griff Whalen for one of the final receiving spots on the team. Brazill had 12 receptions for 161 yards and two touchdowns in 2013.
Brazill's suspension adds to what has been an embarrassing offseason for the Colts.
Owner Jim Irsay was arrested and charged with two misdemeanors for driving under the influence of prescription drugs in March. He has a July 28 pre-trial hearing and an Aug. 28 trial date. Outside linebacker Robert Mathis, the league leader in sacks last season, was suspended the first four games of the 2014 season in May for violating the league's substance-abuse policy.