Jason La Canfora of CBS Sports reported there's "a lot of buzz" linking Rudolph to L.A.
The 22-year-old South Carolina native was the only one of the class' six prominent quarterbacks not to land in the first round after the Baltimore Ravens traded up to select 2016 Heisman Trophy winner Lamar Jackson at No. 32 to close out Thursday's proceedings.
Rudolph completed 65 percent of his throws for 4,904 yards with 37 touchdowns and nine interceptions during his senior season with the Cowboys. He added 10 rushing scores.
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In January, he expressed confidence in his ability compared to the more hyped members of the QB class during an appearance on The Jim Rome Show, per Joe Buettner of Diehards.
"I think I'm the best quarterback in this class, for sure," Rudolph said. "You look at the tape and you look at what I've done over four years, the experience I've had, the starts, the ability to push the ball down the field and athletic enough…more athletic than people give me the credit for and get out of situations."
Philip Rivers is the Chargers' undisputed starter under center, and that won't change even if the front office does move up to grab Rudolph. But Rivers is 36 and a potential free agent after the 2019 season, so L.A. would be wise to start planning for the future.
The Chargers' next scheduled selection comes at No. 48 overall, the 16th pick of the second round. Moving ahead of the Denver Broncos at No. 40 could be enough to land the Oklahoma State standout.