Cole was waived by the Patriots on Dec. 26 after playing in 13 games for them this season. A five-year veteran, his primary contributions came on special teams.
On defense, the 5-foot-10, 195-pound Cole is best suited to play the slot. He totaled 15 tackles and one interception this season while being charted on the field for 167 snaps.
After playing with the New York Jets for the first three years of his career (2009-11), Cole signed a free-agent deal with the Patriots in 2012 and played in 14 games (one start) that year.
The Broncos placed starting cornerback Chris Harris on season-ending injured reserve Monday.
In other injury news, the Patriots believe there is a possibility that punter Ryan Allen will be able to play Sunday, depending on how his shoulder responds to aggressive treatment this week, according to sources.
The plan with Allen, while fluid, reflects why the Patriots have not had any free-agent punters in for workouts as of 4 p.m. ET on Tuesday.
Information from ESPN.com's Patriots reporter Mike Reiss was used in this report.