If he’s not the top pitching Trevor in this free-agent market — or perhaps not even the second-best such entity — Trevor May has performed very well out of the bullpen the past three seasons. Now he holds a place in Mets history unlike any player, first names notwithstanding.

The 31-year-old right-hander is Steve Cohen’s first significant Mets import.

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An industry source confirmed that the Mets and May, a 31-year-old right-handed reliever, have agreed to a two-year contract, pending a physical. SNY first reported the agreement after MLB.com first reported that the two sides were deep into talks.

As long as he clears medically, May, who missed all of 2017 due to Tommy John surgery on his pitching elbow, will become a top setup option for Mets manager Luis Rojas. His arrival will also increase the likelihood of Seth Lugo, who switched from the bullpen to the starting rotation in the middle of last season, remains a starter. The May signing also could impact some calls the Mets must make on Wednesday’s deadline to tender contracts to players under team control.

With the Twins in 2020 during the COVID-shortened season, May struck out 38 batters and walked only seven in 24 games (23 ⅓ innings) while recording a 3.86 ERA. From 2018 through 2020, May totaled a 3.19 ERA with Minnesota, fanning 153 and walking 38, in 113 games totaling 113 innings. That strikeout rate of 32.9 percent placed him 15th in the majors for those clocking at least 100 games in that time period.

“He’s filthy,” an American League Central executive said of May. “31, but backend guys who were starters are really intriguing. Most have multiple pitches, can set up hitters and have some command. Take that and climbing velocity and you have Wade Davis-like stuff. Keep him out of the Central.”

May’s average fastball velocity has risen from 94 miles per hour in 2018 to 95.5 in 2019 to 96.3 in 2020. One notable negative: He did allow five homers last season, giving him a career-worst 1.9 homers per nine innings rate. Like all 2020 stats, you shrug off some of that due to the small sample.

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He’d join a Mets bullpen featuring closer Edwin Diaz, who will be tendered a contract, as well as signed right-handers Jacob Barnes, Dellin Betances, Brad Brach and Jeurys Familia. The Mets face tougher tender/non-tender calls Wednesday on relievers Miguel Castro, Robert Gsellman and Chasen Shreve.

The Mets remain candidates to sign Trevor Bauer, the top available free-agent starting pitcher, after Marcus Stroman accepted the team’s one-year, $18.9-million qualifying offer to bolster the rotation. Veteran closer Trevor Rosenthal, who enjoyed a bounce-back 2020 with the Royals and Padres, also is a free agent.