Barnes, 40, said his championship ring is in the office of Raymond Ridder, the Golden State Warriors' vice president of communications.
The Warriors signed Barnes after All-Star Kevin Durant sustained a hyperextended left knee injury. Barnes averaged 5.7 points, 4.6 rebounds and 2.3 assists per game in 20 appearances for the Warriors during the regular season before he logged 61 minutes in 12 post-season games.
He scored 9 points and had 9 rebounds, 7 assists and 2 steals during the Warriors' championship run.
"The journey is what makes it. I didn't sweat ... I didn't get to guard LeBron [James] or Kawhi [Leonard] so I didn't count my [championship]," Barnes said.
Barnes also said he sustained a sprained ankle that post-season and wasn't healthy until the end of the second round of the playoffs.
"I came in when [Durant] went down, playing a consistent 20 to 25 minutes," Barnes said. "The game KD comes back, I get hurt maybe a week before the playoffs and I'm out of it. I got a free ride, I got a free ring.
"I don't count that as a championship."
The Warriors went 67-15 during the 2016-17 regular season before they won their second title in three years that post-season. Golden State went on to win back-to-back titles before they lost to the Toronto Raptors during the 2018-19 post-season. The Warriors now own the worst record in the NBA (15-50) while the league is suspended due to the coronavirus pandemic.