Update: It's reportedly happening
Per source, Eric Bieniemy has indeed been hired as running backs coach in Kansas City.— ProFootballTalk (@ProFootballTalk) January 11, 2013
Bieniemy was Colorado's offensive coordinator the last two seasons so that's your Denver connection. Another connection is the great 5th down game in 1990. Reid was on Missouri's staff for that game while Bieniemy was Colorado's running back.
Also important: Colorado's offense was awful the last two seasons.
Bieniemy was originally a 1991 second round pick by the Chargers, spending four years in San Diego before moving onto Cincinnati, where he played for the Bengals for four years. He ended his career in Philadelphia in 1999, Andy Reid's first year coaching the Eagles.
Bieniemy also worked under Brad Childress all five seasons he was the head coach of the Minnesota Vikings. Childress was in Kansas City interviewing this week although it's unclear what position, if any, he'll have. I'm taking Bieniemy's inclusion in the coaching search as a sign that Childress is probably going to stick around.
The Chiefs current running backs coach is Maurice Carthon, who came over with Todd Haley in 2009.