Memphis has been "intensifying talks" to move the veteran, and he could be dealt ahead of Thursday's NBA draft.
Conley has two years and $67 million remaining on his current contract. He could become an unrestricted free agent in 2020 if he exercises his early termination option.
The Grizzlies finished 33-49 in 2018-19 and might struggle to compete for a playoff spot next year, so they may not be keen on paying Conley $30-plus million. They're also well-positioned to draft his replacement with the No. 2 overall pick in Murray State point guard Ja Morant.
Trading Conley could open more playing time and a bigger role right out of the gate for Morant.
Meanwhile, the Jazz may be looking for an upgrade at point guard, as they seemingly plan to move on from free agent Ricky Rubio.
"[Utah has] let me know that I am not priority No. 1," Rubio said on Catalunya Radio (via Eric Walden of the Salt Lake Tribune). "Then, we have to look for other options."
Conley, who averaged a career-high 21.1 points per game in 2018-19, would likely fit well next to Donovan Mitchell in what would be an exciting Jazz backcourt.
Utah has the No. 23 pick in the 2019 draft, which could be used as the start of a potential trade package with Memphis.