McGlynn, who was released by Washington on Tuesday, played for Chiefs coach Andy Reid with the Philadelphia Eagles from 2008-10. He spent one year with the Cincinnati Bengals and the past two seasons with the Indianapolis colts.
The former fourth-round draft pick has started 48 of the 57 games he's played in the NFL.
The Chiefs are in need of a veteran offensive lineman. Right tackle Donald Stephenson is suspended for the first four games of the regular season for violating the league's performance-enhancing drugs policy. Several other players have been dealing with illness and injury.