TORONTO -- The Dodgers and Brewers have remained in contact about a possible trade involving Ryan Braun , sources say, and it is not difficult to see why.

The Dodgers had a Major League-low .623 OPS against left-handed pitching last year, when they nearly acquired Braun at the non-waiver Trade Deadline. They haven't been much better (.625) thus far in 2017. And Braun has a career 1.029 OPS against southpaws.

Braun, 33, is a good fit for the Dodgers' archrival, too: The Giants have the Majors' worst production -- a .201 OPS -- in left field, where Braun plays every day for the Brewers, who blanked the Blue Jays, 2-0, on Wednesday night.

By the terms of Braun's no-trade clause, the Dodgers are one of six teams that can acquire him without his permission. That is due to change on May 24, when Braun will attain 10 years of Major League service -- and the complete no-trade protection that comes with 10-and-5 rights.

So in one sense, Braun has two trade deadlines this year: May 24, before which teams have greater influence on his whereabouts; and July 31, along with every other player in the Majors.

"Last year, [the no-trade clause] was relatively significant, because the trade almost happened and there were conversations about how we would address that," Braun told MLB.com on Wednesday. "Other than that, I don't think it's really going to be much of a factor at all. I have such a great relationship with [Brewers owner] Mark Attanasio, [general manager] David Stearns and [assistant general manager] Matt Arnold. We're all really on the same page when it comes to my situation.

"Obviously, things can change. Circumstances can change. Who knows where the future takes us? But I don't foresee [the no-trade protection] having much of an impact on anything moving forward, just because of my relationship with Mark, David and Matt."

In other words: Braun knows the type of talent the Brewers must receive if he's going to be dealt. Milwaukee, in turn, is aware of the other teams for whom Braun is willing to play. (While Braun didn't discuss specific possibilities, it's believed that he's comfortable with the idea of playing near his Los Angeles home.)

In Braun's conversations with Brewers management, there's a mutual understanding that neither party will surprise the other.

"It's something I greatly appreciate," Braun said. "It's not something they needed to do, by any means. It's something that meant a lot to me. It was a unique position to be in.

"More than anything, it gives me perspective on what everybody else deals with, [players] who've been through it multiple times and had no idea it was coming. I couldn't imagine -- for other guys who have families, who have wives, who have children in schools -- having to uplift their lives and move somewhere else ... It gave me greater appreciation of how fortunate I've been to spend my first 11 [seasons] in the big leagues with the same organization."

Braun is a veteran voice in Milwaukee's clubhouse now, far removed from his triumphant and turbulent three-year period earlier this decade: He won the 2011 National League Most Valuable Player Award, finished second one year later, and was suspended for the final 65 games of the 2013 season for violations of Major League Baseball's Joint Drug Prevention and Treatment Program.

Although he was an All-Star as recently as 2015, Braun is no longer one of the game's most-talked-about figures. (The Brewers' absence from the postseason since 2011 is one reason why.) Now a father of two young children, he has steadily, if quietly, regained his status as one of the Majors' most productive players.

Only two outfielders have hit 75 or more home runs, with 50 or more stolen bases, since the beginning of the 2014 season: Michael Trout and Braun.

Braun has averaged 25 home runs over the past three seasons, with an .844 OPS that ranks 19th among Major Leaguers appearing in at least 400 games during that span.

"One of the biggest challenges in this game is consistency; the other one is longevity," Braun said. "Those are the two things I take the most pride in. Obviously, my career hasn't gone exactly the way I would have hoped, but I don't have the ability to go back and change anything. The goal is to continue to be the best player I can be, year in and year out.

"I feel like I'm, right now, as good as I've ever been at baseball. I feel like I'm as athletic as I've ever been. But it is definitely different at 33 than it was at 23, so it's a matter of being able to make adjustments to put myself in a position to continue to be one of the best players in the league."

With the 10th anniversary of his debut approaching, Braun smiled Wednesday as he confirmed one story -- related by a former teammate -- about his first Major League game: After finishing 1-for-4 with a double and two RBIs, Braun cheekily apologized in the clubhouse for having only "travel-day bat speed" following his journey from the Minors.

Brewers veterans might've raised an eyebrow at the rookie's self-assurance, but at least it was justified: After a good night's rest, Braun went 3-for-4 with a home run the next day.

"Obviously, I've always been confident in my abilities," Braun said with a grin. "I've always expected myself to have success. It was something said sarcastically ... but with a hint of seriousness."

In many ways, Braun's never lost that swagger -- which makes him all the more appealing to the Dodgers, Giants and other clubs. He believes he could play shortstop in the Majors right now -- "Absolutely," he affirmed -- and is willing to try right field or first base, if that's what his team needs, even though he's most comfortable on the left side of the diamond.

The willingness to move positions is the sign of a good teammate -- and should help Braun's market value, given any new club would need to find a position for him through at least 2020. (Including the current season, he has four years and $76 million left on his contract.)

As the Brewers rebuild -- overshadowed by their World Series-winning neighbors to the south -- the discussion about where Braun will play those remaining seasons is sure to intensify over the coming weeks. And, sure, he admits to wondering if that might be a little closer to his California home.

"Of course," Braun acknowledged. "But at the same time, I always try to stay in the moment ... The more life experience you have, the easier that becomes, the more you realize you should just focus on the things you can control. That's what allows you to be happy and productive and successful."

Braun can be described that way, in what could be his final weeks before returning to the spotlight of a pennant race.