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BOSTON MA - October 11:  Eduardo Rodriguez #57 of the Boston Red Sox holds up his hat after Hunter Renfroe #10 catches Randy Arozarena of the Tampa Bay Rays line out during the first inning of Game 4 of the ALDS at Fenway Park on October 11, 2021 in Boston, Massachusetts.  (Staff Photo By Matt Stone/MediaNews Group/Boston Herald)
BOSTON MA – October 11: Eduardo Rodriguez #57 of the Boston Red Sox holds up his hat after Hunter Renfroe #10 catches Randy Arozarena of the Tampa Bay Rays line out during the first inning of Game 4 of the ALDS at Fenway Park on October 11, 2021 in Boston, Massachusetts. (Staff Photo By Matt Stone/MediaNews Group/Boston Herald)

This is Eduardo Rodriguez’s chance.

Manager Alex Cora is giving him the ball again in a crucial matchup with the Astros for Game 3 of the American League Championship Series on Monday with a chance to make his name one that Red Sox fans won’t forget for a long time.

And it’s a chance for him to make perhaps his final pitch to the Sox’ front office, which will have to examine his candidacy for a contract extension with exclusive negotiating rights in a five-day window following this year’s World Series.

If they don’t come to an agreement, Rodriguez will be a free agent for the first time in his career.

Rodriguez, relievers Adam Ottavino and Hansel Robles, and first baseman Kyle Schwarber are among the Red Sox players eligible for free agency after the postseason.

“Probably towards the end of the season I guarantee you they were thinking about stuff like that,” Cora said Sunday. “But now it’s not even on their radar, honestly. I think what they think right now is how we’re going to try to beat them tomorrow, how I’m going to stay healthy, what I have to do to be ready for tomorrow. But (free agency) doesn’t come up right now. It doesn’t.”

If someone had to guess whether or not Rodriguez would end up back in Boston next year, it’d have to be a coin flip.

The Red Sox won’t be desperate for his services.

They have Chris Sale and Nathan Eovaldi leading the staff. They hold contract options on Garrett Richards ($10 million) and Martin Perez ($6 million), though both should be declined. They have two standout rookies in Tanner Houck and Garrett Whitlock, who are expected to compete for spots in the rotation next spring. And they have two prospects, Bryan Mata and Thad Ward, likely to return from Tommy John surgery at some point in the middle of the season. Connor Seabold and Jay Groome are a pair of young, intriguing possibilities as well.

The idea of Sale, Eovaldi, Houck and Whitlock making up four of the five starters to begin the year seems like a tantalizing one, and a likely one.

“Whitlock, from day one, he has been impressive, and he really hasn’t waned at all,” Ottavino said. “He has been awesome the whole year. I just think he is going to be truly special for his whole career. It’s been awesome to see him from day one, and Tanner too. He has unbelievable stuff.

“I think they both are starters in the future. But this year they’re kind of our super weapons in the ‘pen. It’s great to be at a point in my career where I can be so pumped to be a part of it and help those guys on their journey as they’re starting out.”

Still, the Sox will need to add another starter or two before 2022.

Rodriguez probably didn’t make himself a lot of money with his regular season performance in 2021. He finished the year with 157 2/3 innings, 185 strikeouts, 47 walks, 19 homers allowed, a 1.39 WHIP and a 4.74 ERA.

Most of those numbers put him toward the bottom of the league among qualified starters, though his strikeout rate was elite, and the best of his career.

Importantly, he made 31 starts one year after a case of COVID-19 left him with myocarditis and bedridden for most of the 2020 season, which he sat out.

“If you ask me that —  that shows you everything,” Rodriguez said. “I’m 100%.”

He said he was thankful just to be alive after his difficult bout with COVID-19.

“I just want everybody to be safe because I know how it is,” he said. “I know how it feels, and I just got to send thanks to God to be here because a lot of people are dead already because of that, and I just am glad to be here.”

In normal times, a lefty who throws in the mid-90s with an elite changeup and a career 4.16 ERA would surely command $12 million to $15 million a year in free agency. After a down year, and amidst an uncertain market, that could be difficult.

The Sox will have some money off the books in 2022, but chief baseball officer Chaim Bloom has shown a tendency to prioritize bargain deals.

He’s also shown a tendency to prioritize players with proven track records in the postseason, which Rodriguez has a chance to cement with a strong start against the Astros on Monday.

Until his last start against the Rays, he’d been a postseason failure, with 11 runs allowed in 11 2/3 innings.

The Astros knocked him around good in a pair of starts in late May and early June, scoring 12 runs off him in 9 1/3 innings.

“At that point he wasn’t on point,” Cora said. “I think the change-up wasn’t good. It was a bad sinker. … The pitcher that we have now compared to who he was when we went there and then he pitched here, it’s a lot different. The velocity is up. The change-up is where he wants it.”

A great start will mean a lot for the Red Sox. And it’d mean a lot for Rodriguez’s future, either in Boston or somewhere else.