Update: The supplemental draft is over. The Broncos didn't use a pick, and neither did anyone else.
According to NFL Network's Ian Rapoport, the NFL will hold its 2014 supplemental draft today (Thursday) at 1 p.m. ET. Four players are eligible to be drafted this year, also per Rapoport.
Players in tomorrow’s Supplemental Draft: N Mex WR Chase Clayton, UNC LB Darius Lipford, Va.-Lynch DT Lakendrick Ross & SMU RB Traylon Shead— Ian Rapoport (@RapSheet) July 9, 2014
For those who need a refresher on what exactly the supplemental draft is, here's Joel Thorman's explanation:
The NFL supplemental draft operates differently from the regular draft in April. It's designed for those players whose status has changed since the deadline to apply for April's draft. Usually, this means players who, after the deadline to apply for the draft, have been kicked off the team or ruled academically ineligible. It's an opportunity for them to get into the NFL as opposed to waiting an entire year. (...)
Despite popular belief, the supplemental draft order is not the same as the NFL draft order. Basically, teams are divided into three groups and there's a weighted lottery to determine the order. (...)
The draft will be conducted via email and it's sort of like a silent auction. Teams can submit a bid -- a draft pick -- on a player and the highest bid wins the rights to that player. That team will then lose the corresponding pick in the  NFL draft.
In short, teams have the opportunity to use a future draft pick on players eligible in the supplemental draft that were not eligible to be drafted earlier in the year, often players that were in poor standing with the NCAA. Although it is expected that all 32 teams will choose to pass on the four prospects this year, a few notable players have been selected through the supplemental draft in the past.
Quarterback Bernie Kosar (Browns, 1985), wide receiver Cris Carter (Eagles, 1987), running back Bobby Humphrey (Broncos, 1989), defensive tackle Jamal Williams (Chargers, 1998), offensive tackle Jared Gaither (Ravens, 2007), quarterback Terrelle Pryor (Raiders, 2011), and receiver Josh Gordon (Browns, 2012) were all supplemental picks.
Will the Broncos use a supplemental draft pick?
It's unlikely, as Scotty breaks down nicely in this morning's Horse Tracks. Only Darius Lipford, North Carolina linebacker would possibly interested the Broncos, Scotty writes, but even then it's unlikely. The Broncos are an extremely deep team and may have trouble keeping all of their draft picks out of training camp as it is.
For more information on the four players eligible and the players eligible, check out Mocking the Draft's excellent preview.
Six players were eligible in 2013, though none were selected.