Bergstrom was the first pick in the draft for the Raiders, who previously traded their first- and second-round picks for quarterback Carson Palmer and used a third-round pick in last year's supplemental draft on quarterback Terrelle Pryor.
Other Utah players who could be selected include offensive lineman John Cullen and defensive back Conroy Black.
Black ran a 4.3 in the 40-yard dash at Utah's Pro Day and did well in agility drills.
If the 6-foot, 186-pound Black is drafted, he would maintain Utah's streak of sending defensive backs to the NFL.
Brandon Burton was drafted in the fifth round last year by Minnesota, R.J. Stanford was taken in the fifth round in 2010 by Carolina, and Robert Johnson went in the same round to Tennessee. Brice McCain was taken in the sixth round in 2009 by Houston, and Sean Smith was taken in the second round by Miami the same year.
Utah coach Kyle Whittingham said the Utes' success in cultivating defensive talent is in Utah's recruiting and the way the Utes play.
"We recruit a lot of speed, and we play man coverage," he said. "NFL scouts like corners with man-coverage skills. They are a premium in the NFL."
Black earned honorable mention honors in the Pac-12 after he tied for most interceptions with four. He also had 39 tackles and four pass breakups.