According to ESPN.com's Adrian Wojnarowski, the Spurs declined Boston's advances and did not make a counteroffer.
A nagging quad injury limited Leonard, who finish third in MVP voting in 2016-17, to just nine games last season. His health was a topic of conversation throughout the season.
Per Wojnarowski, the uncertainty regarding Leonard's status led to a strained relationship between Leonard and his coaches and teammates.
Leonard and head coach Gregg Popovich are reportedly scheduled to meet soon to discuss his long-term future with the organization and whether the Spurs are willing to offer him a supermax contract extension (five years, $219 million). If Leonard chooses to opt out, he can become a free agent after the 2018-19 season.
Boston reached the Eastern Conference Finals despite being without forward Gordon Hayward for all but one game and guard Kyrie Irving for the entire playoffs. Provided the Spurs change their tune on trading Leonard, Boston would likely be the most logical trade partner considering all it could offer.
The Celtics are stacked with young, talented players such as Jayson Tatum, Jaylen Brown and Terry Rozier, and they own the Sacramento Kings' 2019 first-round pick, which is protected only if it is No. 1 overall.
L.A. and Philadelphia have some up-and-coming stars of their own. The Lakers boast the likes of Brandon Ingram, Kyle Kuzma, Julius Randle (restricted free agent) and Lonzo Ball, while the Sixers have Ben Simmons, Joel Embiid, Markelle Fultz and Dario Saric, although it is difficult to envision Simmons or Embiid going anywhere.
There will likely be some enticing offers for the Spurs to consider, but Leonard may have proved just how valuable he was last season by barely playing. The Spurs went 47-35, their first season with fewer than 50 wins since the 1998-99 lockout-shortened campaign.
The Golden State Warriors also handled them quite easily in the first round of the playoffs, winning 4-1.
Among players who appeared in at least 10 games, All-Star LaMarcus Aldridge was San Antonio's leading scorer at 23.1 points per game. Rudy Gay was second at just 11.5 points per contest.
Leonard, meanwhile, is a two-time All-Star, two-time NBA Defensive Player of the Year and an NBA Finals MVP.