According to Adrian Wojnarowski of ESPN (via Jeff Garcia of Fox San Antonio), "a lot of teams" are interested in a trade, but talks are "not being entertained yet by the Spurs."
Leonard is under contract for next season with a player option for 2019-20.
The 26-year-old played only nine games last season due to a quadriceps injury, and there was never a clear timeline for his return.
Ramona Shelburne and Michael C. Wright of ESPN.com reported there's "tension" between player and team with a "looming showdown," as the Spurs believe Leonard and his group have "an ulterior motive to fray the relationship and get Leonard traded to a larger market."
Rumors have since run wild on a potential trade, although Kevin O'Connor of The Ringer noted the organization "won't settle for anything less than a grand-slam offer."
Meanwhile, there is no certainty the Spurs will make a deal at all.
Tom Osborn of the San Antonio Express-News reported the front office hopes to mend fences and offer Leonard a supermax deal worth $219 million over five years.
Considering the forward finished in the top three of MVP voting in each of his last two healthy seasons, there is plenty of motivation to keep him on the roster. Even if the relationship has been strained, it's clear the Spurs aren't ready to give up on him.