An officer with the Arizona Department of Public Safety stopped Awasom's vehicle at about 3 a.m. Friday on Interstate 10 in Phoenix, said Sgt. Tim Mason.
Blood test results aren't yet available, Mason said, but Awasom was cited on suspicion of driving under the influence and extreme DUI and was released.
Awasom, who arrived Thursday for Sunday's Super Bowl, has been on injured reserve since Sept. 18.
Giants spokesman Pat Hanlon said the team was aware of the situation but would not confirm any details.
Awasom was sent home by the team, Hanlon said.
In his third year in the league, Awasom played in the first two games this season, injuring his neck in the second game against Green Bay.
He was signed as a free agent in May 2005 and played in five games that season, mostly on special teams.
Awasom was waived before the start of the 2006 season and then signed to the practice squad. He was signed to the 53-man roster Oct. 26, 2006, and played in 10 games, finishing with 10 tackles, a sack, five quarterback hurries and four special teams tackles.
