Mincey is expected to be formally signed and on the practice field Wednesday. He was released by the Jacksonville Jaguars on Friday after being late to a team meeting.
The eighth-year veteran played for Broncos defensive coordinator Jack Del Rio when Del Rio was the coach of the Jaguars.
Mincey also had been left off a road trip to Houston last month after he was late for a team meeting, and he was a game-day inactive against the Cleveland Browns on Dec. 1.
The Broncos talked to Jaguars general manager David Caldwell before coming to terms with Mincey and hope Mincey can enter their rotation up front. They will have to make a roster move to formally add him.
Derek Wolfe has not practiced since suffering what the Broncos called "seizure-like symptoms" on the team's Nov. 29 bus ride to the airport for the trip to Kansas City.
The Broncos could move Wolfe to injured reserve, but as of Monday team officials were still hopeful he could return for the postseason. Wolfe recently began working with strength and conditioning coaches.
Mincey played well enough at times for Jacksonville to have signed a four-year, $20 million contract last year. The deal included $9 million guaranteed.
He had a career-high eight sacks in 2011. The number dropped to three in 2012, and he had 14 tackles and two sacks in eight games this season (he also missed three games with a concussion).
Mincey turned 30 on Saturday and Denver hopes the lure of the playoff chase helps him to refocus.
To make room for Mincey, the Broncos waived center J.D. Walton.
Walton started the season on PUP list after offseason ankle surgery to deal with an infection -- he originally fractured the ankle in the fourth game of the 2012 season but had been practicing with the team of late. He was a third-round pick by the Broncos in the 2010 draft and had played every snap in his career until his injury last season.