According to a copy of the police report, Smith was attempting to make a turn in Kansas City's Power & Light entertainment district when he struck the light pole around 12:30 a.m. The officer reported that Smith "appeared confused and began mumbling statements that I did not understand."
Smith declined medical attention. Along with citations for driving under the influence, he was also cited for failure to provide insurance.
The Chiefs said in a statement on Tuesday that they were aware of the matter. The statement called the situation "disappointing" but declined further comment.
The 26-year-old Smith signed a three-year, $16.5 million contract with the Chiefs as a free agent last offseason and played in 16 games (15 starts), posting 47 tackles with two interceptions and 16 passes defensed.
The Kansas City Star reported that Smith was at practice Tuesday but worked mostly with the Chiefs' second-team defense as Ron Parker, a backup in 2013, got most of the first-team reps.
The Associated Press contributed to this report.