The deal is worth $5.25 million and Smyly could earn an additional $2.5 million in bonuses, a source told ESPN's Jesse Rogers.
Smyly, 32, won a World Series with the Atlanta Braves last season, when he went 11-4 with a 4.48 ERA in 29 appearances (23 starts). He pitched twice in the Braves' World Series victory over the Houston Astros, allowing three runs over four innings.
He signed a two-year, $10 million contract with the Cubs in December 2017 but never pitched for Chicago. Working his way back after undergoing Tommy John surgery in 2017, Smyly ran out of time to get into a big league game in 2018 and was traded to the Texas Rangers that November.
Villar is slated to make $4.5 million in salary in 2022, and there is a $10 million mutual option with a $1.5 million buyout. He earns a $500,000 performance bonus for reaching 500 plate appearances. Norris' contract is worth $1.75 million, and the lefty can earn up to $2 million in incentives. Norris played for Detroit and Milwaukee last year, going 2-3 with a 6.16 ERA in 56 games.
The Associated Press contributed to this report.