Former Baylor wide receiver Josh Gordon is widely considered the best available prospect in this year's supplemental draft. Gordon declared for the draft one year after being dismissed from the Baylor football team (transferred to Utah, but sat out the year due to eligibility ratings).
At 6'3" and 225 lbs, Gordon is a big, physical specimen who put up big time numbers his sophomore season at Baylor. One scout went as far as to tell ESPN that he was a "rare" talent, and some have even said he could go as high as the second or third round.
While the Patriots have yet to select a player through the Supplemental Draft since Bill Belichick took over the team in 2000, if the right prospect were to come along at an appropriate draft pick compensation, I don't think the Patriots would hesitate to pull the trigger.
That being said, of all the supplemental draft prospects that have come through over the years, Josh Gordon is certainly one of the most intriguing ones in recent memory. The Patriots have a rapidly aging receiving corps, and could use a big, athletic body at the position. While it is unlikely that the Patriots will put in a claim on Gordon, I wouldn't rule it out entirely.
In fact, the Patriots are one of a large group of teams expressing some level of interest in Gordon. According to Adam Caplan, the Patriots were in attendance at Gordon's personal Pro Day today. Now, that doesn't really give an indication of exactly how interested the Patriots are, but it's worth noting, and certainly adds more intrigue for Patriots fans to July 15th's Supplemental Draft.