The Spurs formally informed the Grizzlies of their decision Sunday night, shortly before a midnight ET deadline, a league source confirmed to ESPN's Adrian Wojnarowski.
Anderson, 24, was a primary target of the Grizzlies and is likely to become their starting small forward. He has had an outstanding plus-minus on defense the past two years.
Memphis considers the offer sheet to Anderson an investment in the franchise's stated goal of recapturing the "Grit and Grind" culture under new coach J.B. Bickerstaff.
Anderson averaged 7.9 points and 5.4 rebounds in 26.7 minutes per game last season for the Spurs, who drafted Anderson with the 30th pick in the 2014 NBA draft, after he left UCLA following his freshman season.