MILWAUKEE — A funny thing happened on Aaron Judge’s way to hitting a ton of home runs: a ballooning batting average and a second, different chase at history.

Judge has spent most of the season on pace to threaten Roger Maris’s American League record of 61 home runs in a season, set in 1961. With two home runs in Sunday’s 12-8 win over the Milwaukee Brewers, Judge’s season total reached 59 — 20 more than any other batter in the major leagues.

But as Judge, 30, has racked up the home runs in the second half of the season — 26 since the All-Star break in mid July — he has crept up the A.L.’s batting leaderboard as well. With a torrid .491 batting clip (27-for-55) this month, Judge has put himself in contention to claim a triple crown, an impressive feat in which a batter leads his league in batting average, home runs and runs batted in.

With two and a half weeks left in the regular season, Judge holds huge leads in both home runs and R.B.I. The closest A.L. batter in home runs is Houston’s Yordan Alvarez, with 37 — 22 behind Judge — while Cleveland’s José Ramírez is second in R.B.I., but his 112 are 15 behind Judge.

After collecting four hits on Sunday, Judge was closing in the third triple crown category. His .316 average was tied with Boston’s Xander Bogaerts for second in the A.L., trailing only Minnesota’s Luis Arráez, at .317.

“That’s remarkable,” said Yankees ace Gerrit Cole of Judge, adding later, “It’s the greatest offensive season that I’ve personally ever witnessed. I don’t know what else to say. It’s wonderful.”

A triple crown is such a rare achievement in the American and National leagues that a group of only 10 players accounts for the 12 triple crowns. The last person to accomplish the feat — Miguel Cabrera, in 2012 — was the first to do it in over four decades. Yet this year, both leagues have the potential to add another to the list: Judge and, less so, Paul Goldschmidt of the St. Louis Cardinals, who is second in the N.L. in batting average (.322), tied for third in home runs (35) and second in R.B.I. (112).

“He’s really just amazing,” said Yankees first baseman Anthony Rizzo of Judge. Alluding to Cabrera, Rizzo added, “Miggy was the last one to do it and the numbers he put up were astronomical and Aaron is blowing that away right now. We haven’t seen it in this game in a long time and for him to be doing it during this era of baseball is just incredible.”

Judge’s previous career high in batting average was .287 and he had hit higher than .280 only twice in his six seasons. Yankees Manager Aaron Boone said he had never given much thought to Judge winning a batting title, but he knew that Judge took a lot of pride in hitting for a high average. He added, “So when you consider what he’s doing and the person and his talent, nothing is surprising.”

Asked if winning a batting title or triple crown were ever among his dreams growing up, Judge smiled and said, “You set personal goals, especially as a 10-year-old kid. You have dreams and aspirations of what you want to try to accomplish in Major League Baseball if you ever get there. But never in your wildest dreams do you think it’d ever come true. You hope, you pray, you work, but you never know if it’s going to happen until it does.”

When Cabrera won the triple crown, Judge was a sophomore at Fresno State University, still a year away from being a first-round pick by the Yankees. His head coach at the time, Mike Batesole, made the entire team watch a YouTube video of all of Cabrera’s 205 hits. “It was a long video,” Judge said, laughing.

“It was well worth the watch,” he continued. “He really hammered home just how simple he makes everything. He doesn’t try to overswing. He doesn’t try to do too much, especially with guys on base, you see him poke a single to right field and score two runs. He did the little things in the game and you do that over 162, you’re going to have a pretty good year.”

That’s exactly what Judge did in the ninth inning on Sunday. Earlier in the game, he had helped the Yankees come back from a 4-1 deficit. In the third, Judge blasted a sinker from Brewers starter Jason Alexander 414 feet into the second deck in right field. Four innings later, Judge hit the ball even farther, 443 feet, but to the second deck in left field. Perhaps exasperated, Brewers reliever Luis Perdomo threw his arms up just after Judge made contact.

When the Brewers tightened things up late, Judge delivered yet again. As he strode to the plate in the ninth inning, Yankees fans at American Family Field chanted “M-V-P!” Judge clobbered the first pitch he saw from side-arming reliever Trevor Kelley toward left field. The crowd cheered then sighed and clapped when the ball bounced in front of the wall for a two-run double. It was the final stroke in a 4-for-5 day, which included a walk and four R.B.I.

Afterward, Judge said he wasn’t thinking about a home run but rather driving in more runs. No. 60 will have to wait for another day — the Yankees next play on Tuesday, at home — but Judge, after all, has more than one run at the record books underway.

“I understand there’s going to be a story line for the next two weeks on who deserves the M.V.P.,” Rizzo said, then alluding to steroid testing that began in M.L.B. 20 years ago. “But this hasn’t been done in this era, someone chasing 61 clean like this and no real question marks on what’s going on in the game.”

As he continued making the case for his teammate, Rizzo referenced Anaheim’s Shohei Ohtani, the two-way star who unanimously won the A.L. Most Valuable Player Award last year and has been even better on the mound this season. “So it’s really impressive. It’s really fun. I know there’s a debate about Ohtani and whatnot, but Judge, it’s his season. He’s having a special season.