Gay agreed to return to San Antonio after last season's first-round playoff exit, he told The Undefeated's Marc J. Spears on Sunday.
"Ready to compete for a 'ship. Last season wasn't how we wanted to end things. We will be better next season," Gay said.
According to multiple reports, Gay agreed to a two-year, $32 million deal.
There were questions about what the future would hold for Gay after he tore his Achilles tendon while a member of the Sacramento Kings in 2017, but the veteran forward has provided plenty of answers since.
He signed with the Spurs in July 2017, going on to appear in 57 games once healthy and averaging 13.7 points, 6.8 rebounds and 2.6 assists per game. He opted out of his two-year deal, then re-signed with San Antonio last summer on a one-year, $10 million contract.
Gay's play in 2018-19 earned him another big payday. He appeared in 69 games and made 51 starts while averaging 13.7 points on career bests of 50.4% shooting from the field and 40.2% from 3-point range. He added 6.8 rebounds and 2.6 assists per game.
San Antonio also gave Gay something he lacked before joining it -- playoff experience. Gay had reached the postseason just once (Grizzlies in 2012) prior to signing with the Spurs, but doubled that output the past two seasons.
A few days after the Spurs' season wrapped up in a Game 7 loss to the Nuggets, coach Gregg Popovich said he was hopeful San Antonio would bring back Gay while noting the veteran is "somebody who changes games" with his scoring on the court while being a "great teammate" off it.
"We are hoping we can figure out a way to keep him here," Popovich told reporters in May.
Gay, who turns 33 in August, is entering his 14th NBA season, with previous stops in Memphis, Toronto and Sacramento. The former first-round pick (eighth overall in 2006) out of UConn holds career averages of 17.6 points, 5.9 rebounds, 2.2 assists and 1.3 steals per game.
Carroll announced his move to the Spurs on social media Sunday. Multiple reports said will sign a two-year deal worth between $12-13 million.
"San Antonio Let's Work EVERYDAY," Carroll wrote in a post on his Instagram account. He continued in the caption, "Thank You @spurs for WELCOMING ME."
Carroll, who will turn 33 in July, spent the past two seasons with the Brooklyn Nets, largely coming off the bench last season after starting in 2017-18. He averaged 11.1 points with 5.2 rebounds per game last season.
The Missouri product has played for the Grizzlies, Rockets, Nuggets, Jazz, Hawks and Raptors in his 10-year career, never spending more than two seasons with any one team.