Blevins, 33, gets a $5.5 million salary this year, and the Mets have a $7 million option for 2018 with a $1 million buyout. He can earn $100,000 annually in performance bonuses.
He agreed to the deal last week but needed to pass a physical for the agreement to be finalized.
Right-hander Fernando Salas, another member of the Mets bullpen last year, also agreed to a deal last week that is in the process of being completed.
To clear a spot for Blevins on the 40-man roster, the Mets designated infielder/outfielder Ty Kelly for assignment.
Blevins went 4-2 with a 2.79 ERA and two saves in a career-high 73 appearances last year during his second season with the Mets. He struck out 52 in 42 innings and stranded 47 of 55 inherited runners, holding opponents to a .167 batting average (6 for 36) with runners in scoring position.
Blevins missed all but the first two weeks of the 2015 season after breaking his arm.