Shams Charania of The Athletic and Stadium first reported details of the deal.
Gay bounced back during the 2018-19 season after struggling to find a niche during his first year with the San Antonio Spurs.
The 32-year-old forward averaged 13.7 points, 6.8 rebounds and 2.6 assists across 69 appearances (51 starts). He shot a career-best 50.4 percent from the field, including 40.2 percent from beyond the arc, and 81.6 percent on free throws.
All of those numbers were improvements over the 2017-18 campaign.
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Spurs head coach Gregg Popovich praised the UConn product after the team's season ended with a first-round loss to the Denver Nuggets in the playoffs.
"Rudy is a scorer," he told reporters. "He's somebody who changes games. And he's gotten better rebounding-wise and defensively, which was never his forte. But he understands the importance of it and is just getting better every year."
Popovich added: "He's a great teammate, has a great personality, great sense of humor."
Gay has likely added years to his career and money to his bank account by taking some reliance off his offensive skill by improving his all-around game.
Popovich's ability to create roles for his secondary contributors to succeed is a key reason he's a five-time NBA champion and three-time NBA Coach of the Year. Gay's transformation over the past two years is a perfect example.
In turn, it's no surprise the forward has opted to return to the Spurs, where he should fill pretty much the same role in 2019-20.