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Alex Speier

The Red Sox are keeping their eyes out for the next Ryan Brasier

Relief pitcher Ryan Brasier made 34 of 41 career major-league appearances last season at age 30.
Relief pitcher Ryan Brasier made 34 of 41 career major-league appearances last season at age 30.(Jim Davis/Globe Staff)

The names keep coming off the board. Most of the high-end relief pitcher targets — Zach Britton, Jeurys Familia, Andrew Miller, Adam Ottavino, Joakim Soria, and Kelvin Herrera, as well as 2018 World Series standout Joe Kelly — are gone on multiyear deals. That they didn’t find their way to the Red Sox comes as little surprise, given that the Red Sox made clear early in the offseason that they’d be staying away from big-dollar expenditures on the bullpen.

But more recently, players with closing experience have come off the board for more modest dollar figures, with one-year deals for players such as Cody Allen ($8.5 million from the Angels), Shawn Kelley ($2.75 million with the Rangers), and Greg Holland ($3.25 million from the Diamondbacks). None ended up in Boston.

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The Red Sox have remained content to hunt on the periphery. They traded in December for righthander Colten Brewer; they’ve added players who missed most or all of 2018 on minor league deals in Zach Putnam and Erasmo Ramirez; and most recently, they reached a minor league deal with Jenrry Mejia, a righthander who was once a top Mets prospect and closer but received a lifetime suspension in 2016 for a third positive test for performance-enhancing drugs. Mejia, who was reinstated by the commissioner’s office last year, briefly played with Red Sox manager Alex Cora in New York in 2010.

“Obviously, [Mejia] made some mistakes,” Cora said on Friday at a lunch to benefit the nonprofit Foundation To Be Named Later. “He went to the [Dominican winter league]. He pitched well. He’s going to be in the Caribbean Series in Panama. We’ll see what happens.”

That approach characterizes a lot of what the Red Sox are doing this year. The team hasn’t ruled out signing a reliever to a big league deal — preferably one with a skill set that isn’t yet present among the relievers who are on their roster. So, for instance, Jake Diekman — a hard-throwing lefthanded pitcher with reverse splits (he held righties to a .191/.303/.321 line with a 31.6 percent strikeout rate with the Rangers and Diamondbacks last year) — could be of interest.

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But for now, the team remains focused chiefly on lesser-known pitchers on the market — with interest chiefly in those who are interested in one-year deals for relatively low salaries. Otherwise, the team is content to replace Kelly and closer Craig Kimbrel with the pitchers who they have — while leaving the door open to trades (either before the season or, perhaps more likely, during the year) to round out their late-innings group.

“We’re very comfortable with [the bullpen],” assistant general manager Eddie Romero said. “We’re open-minded to anybody that’s available before spring training starts or even into spring training. We like the group we have. If there’s an obvious upgrade that can help for us, that might fall into a range of interest for us, we’d definitely consider it.

“We’re always looking to make upgrades here and there, but we’re very comfortable with this group. We feel confident that we can mix and match with the players that we have right now . . . I know there are questions about it, but we’re confident.”

Part of that comfort stems from what happened in 2018. In a year where the team faced considerable scrutiny for its relative inactivity on the reliever market at the trade deadline, the emergence of Ryan Brasier — signed as a minor league free agent without an invitation to big league camp — down the stretch gave the team enough depth to win a title.

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Cora was reminded of Brasier’s unlikely path on Friday morning, when he watched a replay of the TBS broadcast of Game 4 of the American League Championship Series against the Astros. Brasier delivered 1⅔ scoreless innings in an 8-6 win. Cora took note of the commentary from TBS analyst Brian Anderson.

“He’s like, ‘Brasier got the last out of the sixth inning. This is the coming-out party of Ryan Brasier.’ We’re looking for a lot of coming-out parties,” Cora said. “We signed a lot of guys throughout the process. We have some capable guys that are coming. It’s not that we’re looking for the next Ryan Brasier — it’s not that easy — but somebody’s going to get a chance to come up and pitch.”

The opportunities will go not to the expected names — Miller or Britton or Familia or Ottavino — but instead in all likelihood to those who enter camp in obscurity and who, the Sox hope, rise from it. As much as Cora might downplay the idea of a search for the next Brasier, the Red Sox are making a considerable bet that it’s possible to repeat, if not the exact success they experienced with him, then something close to it in the making of a bullpen.

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Alex Speier can be reached at alex.speier@globe.com. Follow him on Twitter at @alexspeier.