The demand comes a day after the Jazz traded up in the draft to get University of Michigan star Trey Burke, who was selected ninth on Thursday night.
"He's not going to come back as a backup," the source said of Williams.
As for Burke, he told reporters Thursday, "My goal is to come in and start right away. I look forward to coming in and helping the team win."
Williams, a 10-year NBA veteran, averaged 12.9 points and 6.2 assists during his injury-plagued 2012-13 season in which he played 46 games.
The Jazz were adamant in stating their need for a point guard in this year's draft class. They dealt their two first-round picks (Nos. 14 and 21) to the Minnesota Timberwolves in order to get Burke to Salt Lake City.
Burke averaged 18.6 points and 6.7 rebounds as a sophomore last season while leading Michigan to the NCAA championship game.
More NBA coverage from Yahoo! Sports
• Rebuilding Celtics are big draft day winners
• Celtics-Nets deal a shocker on so many levels
• Anthony Bennett speechless after being picked No. 1
• David Stern trolls fans in final NBA draft
- Sports & Recreation
- Utah Jazz
- Mo Williams