There have been rumors about San Antonio's intention to trade up, but it might not be worth it if it means becoming less competitive in the short term.
The draft is mere hours away and the Spurs keep showing up on rumors about trading up. First Alex Kennedy from Basketball Insiders mentioned it, then NBADraft.net. Recently ESPN's Chad Ford mentioned that San Antonio is looking to get the 13th pick from the Phoenix Suns. There hasn't been this much buzz about the Spurs going into draft day since the 2011, when they did in fact pulled off a move that netted them Kawhi Leonard.