Familia had 123 saves for the Mets, who traded him to Oakland last July. He instructed agents Sam and Seth Levinson to pursue any possibility to return to the Mets.
"Honestly, that was my goal: be back with the Mets," Familia said during a conference call Monday, three days after his $30 million, three-year contract was finalized.
"I want to pitch whatever innings they want to put me in the game," Familia said. "He's one of the best relievers in the game, the best closer. It's going to be fun."
New York also announced Monday that it had agreed to a minor league contract with 38-year-old outfielder Rajai Davis, who will report to major league spring training.
The Mets also called a news conference for Tuesday to introduce catcher Wilson Ramos, who finalized a $19 million, two-year contract. He receives a $2 million signing bonus payable next July 1, $6.25 million in 2019 and $9.25 million in 2020. The Mets have a $10 million option for 2021 with a $1.5 million buyout. Ramos would earn a $500,000 bonus in each season he has 100 starts at catcher.
Familia gets a $2 million signing bonus payable in equal installments on July 1 and Dec. 10 next year. He has salaries of $6 million in 2019 and $11 million in each of the final two seasons. The Mets will defer $1 million of his 2020 salary, payable on Jan. 15, 2022. They will defer the same amount of his 2021 salary, payable on July 1, 2022.
He gets a $500,000 performance bonus in any season in which he finishes 50 games. He would get a $1 million assignment bonus if traded.
A 29-year-old right-hander, Familia said returning to New York was a preference because his wife's family lives in the area.
He was 8-6 with 18 saves in 24 chances and a 3.13 ERA for the Mets and A's last season. He signed with the Mets at age 17 and had 43 saves in 2015 when New York reached the World Series. He led the majors with a franchise-record 51 saves the following year and was an All-Star, again helping New York reach the playoffs.
Familia served a 15-game suspension at the start of 2017 for violating Major League Baseball's domestic violence policy. Later that year, he had a blood clot that required surgery, and he finished 2-2 with a 4.38 ERA and six saves in 26 games.
Last season, he was 4-4 with a 2.88 ERA and 17 saves in 21 chances with the Mets.
Davis hit .224 with one home run, six RBIs and 21 steals in 101 games and 216 plate appearances for Cleveland. His two-run, eighth-inning homer off Aroldis Chapman tied Game 7 of the 2016 World Series for Cleveland against the Chicago Cubs, a game the Indians lost in 10 innings.
Davis has a .262 career average with 61 homers, 379 RBIs and 415 stolen bases in 13 major league seasons.