According to Stein, the Warriors believe their chances of landing the former NBA MVP were enhanced by their NBA finals loss to the Cleveland Cavaliers.
Meanwhile, The Vertical's Adrian Wojnarowski reports that in addition to the Thunder, Warriors, and Spurs, Durant's Roc Nation representatives are also working to cement meetings with the Boston Celtics, Los Angeles Clippers, and Miami Heat.
Durant declined his player option earlier this week, becoming an unrestricted free agent. The Thunder own Durant's Bird rights, meaning they can offer the 27-year-old more years and money than any other team.
Reports surfaced earlier in the year that Durant would only be willing to leave Oklahoma City if he signed with a championship contender.
The 6-foot-9 small forward averaged 28.2 points, 8.2 rebounds, and 5.0 assists on 50.5 percent shooting this season.
Teams cannot meet with Durant until July 1, with Wojnarowski reporting that the meetings will take place in New York. The Warriors are reportedly set to use Steph Curry and Klay Thompson to help lure Durant, while the Spurs will ask Tim Duncan and Gregg Popovich to support them in their pitch, according to Stein.