The newspaper had reported last week that he would be the team's offensive coordinator; however, Norm Chow currently holds that position.
Chow is under contract for $500,000 in each of the next two seasons. However, coach Rick Neuheisel publicly acknowledged that he was considering changing Chow's role.
"The offense is still a place I'm evaluating and Norm as the coordinator is certainly in that category," Neuheisel said on Dec. 18. "But I'm just trying to figure out what is the best way to go, whether it's a reshuffling or if it's new personnel. I'm still sorting that out in my own mind."
The Salt Lake Tribune has reported that Utah is interested in making Chow its offensive coordinator when he reaches some kind of resolution with UCLA.
Johnson was hired as 49ers quarterbacks coach in 2009 and promoted to offensive coordinator last season. He was an assistant with Neuheisel with the Baltimore Ravens.
The Bruins finished 4-8 last season, prompting Neuheisel to fire defensive coordinator Chuck Bullough and wide receivers coach Reggie Moore.