The agreement was first reported by NBA.com.
Barnes posted a farewell to the Warriors on Twitter on Tuesday.
West played in 78 games with 19 starts during the 2015-16 season with 19 starts. He averaged 7.1 points and 4.0 rebounds on 54.4 percent shooting as a key cog on a San Antonio bench that produced an NBA-best plus-10.9 net rating. That rating was the best by a bench since the shortened 2011-12 season, when the Chicago Bulls led the NBA with a plus-12.7 mark.
West, 35, harbored no regrets about turning down a more lucrative financial situation with the Pacers to join the Spurs.
West played just nine minutes during San Antonio's final game of the postseason. He had a $1.5 million player option for the 2016-17 season, which he declined.
ESPN.com's Michael C. Wright contributed to this report.