Last year the supplemental draft came and went without a single player being selected. The NFL's supplemental draft works a bit different than the real draft with teams bidding one of their next season's draft pick for the prospect. If you are awarded a player in the supplemental draft you then forfeit your pick in that same round in the next draft.
Good old Wiki breaks it down like this.
Draft order is determined by a weighted system that is divided into three groupings. First come the teams that had six or fewer wins last season, followed by non-playoff teams that had more than six wins, followed by the 12 playoff teams. In the supplemental draft, a team is not required to use any picks. Instead, if a team wants a player in the supplemental draft, they submit a "bid" to the Commissioner with the round they would pick that player. If no other team places a bid on that player at an earlier spot, the team is awarded the player and has to give up an equivalent pick in the following year's draft.
The Chicago Bears last used the supplemental draft for running back Harvey Unga in 2010, whom they grabbed in the 7th round. Since they nabbed him in the 7th, they lost their 7th rounder in the 2011 NFL Draft.
This year there are four prospects that are hoping their NFL dreams come true.
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
As reported by Draft Diamonds, the Chicago Bears were in attendance for the workout of 6'4", 366 pound defensive lineman Lakendrick Ross formerly of the Virginia University of Lynchburg. He ran his 40 yard dash in the 5.4 range with his 10 yard split around 1.81. For some perspective, Bears rookie DT Will Sutton ran a 5.36 40 with a 1.75 10 yard split. Ross also has a wingspan a half inch shy of seven feet.
Ross was ineligible for the 2013 season and he's considered a very raw prospect. SB nation draft guru Dan Kadar calls him "large and not much else" and "Stiff, slow, stands up off the snap. Long shot at best", and he wouldn't recommend drafting him unless a team has a multiple of 7th round picks. The Bears currently have one 7th round pick in the 2015 draft that they could use to take a flier on Ross. They also have a 6th, but I doubt the Bears would elect to take anyone higher than 7th.
EDIT: Here's some info on the UNC LB.
LB Darius Lipford measured 6-2, 256 at workout. Was to be MLB for Tar Heels as senior, has also played WLB/SLB and "Bandit" at UNC.— Greg Auman (@gregauman) July 9, 2014
EDIT: I didn't see this earlier, but WCG member drocksthaparty posted a Fanpost this morning, and he found a link to the Bears having interest in Shead as well.