Yankees owner Hal Steinbrenner realizes there is a chance the price may be going up to sign Aaron Judge to a long-term contract after the star slugger’s outstanding first half.

Judge turned down an eight-year contract worth $230.5 million to $234.5 million, cutting off talks ahead of the April 8 opener and saying he wouldn’t negotiate again until after the season. Judge’s representatives wanted a nine-year deal in excess of the average annual value of Los Angeles Angels outfielder Mike Trout’s contract, which comes to $319.6 million, a person familiar with the negotiations said then, speaking on condition of anonymity because Judge’s stance was not made public.

“Is it a possibility? Of course it’s a possibility,” Steinbrenner said Wednesday during his midseason media availability. “But cross that bridge when I come to it.”

Judge, eligible for free agency after the World Series, leads the major leagues with 29 homers. He is second in the AL with 60 RBIs and is batting .281, a big reason the Yankees began Wednesday with a major league-best 58-23 record and a 13-game lead in the AL East.

“Aaron is a great Yankee and he’s very valuable to this organization,” Steinbrenner said. “He’s a great leader. Obviously, he’s performed extremely well this year, which we’re all thrilled about. So, look, any great Yankee, yes, it’s something we’re going to be looking at and talking about seriously. It means a lot to the organization. Nobody’s going to deny that.”

After the Yankees made their long-term offer public in April, Judge and the team agreed last month to a $19 million, one-year deal that avoided an arbirtation hearing. Steinbrenner said even if talks do resume during the season, the team wouldn’t publicly discuss negotiations until after the Yankees' season is over.

“We made an offer that I feel was a very good one,” Steinbrenner said. “It was based on the numbers, of course, but it was also based in part as what he means to this organization. It was a combination of both. And we just didn’t get a deal done. We had less time, of course, this offseason than we would normally because of lockout and all. But there’s no regrets. We’re going to be having discussions at some point.”

Steinbrenner expressed satisfaction with the team’s offseason makeover, which included trading catcher Gary Sánchez, acquiring shortstop Isiah Kiner-Falefa, third baseman Josh Donaldson and catcher Jose Trevino, and hiring hitting coach Dillon Lawson. New York didn’t try to sign a premier free-agent shortstop, such as Carlos Correa or Trevor Story.

“To be a great team Gene Michael always used to say defense up the middle, right? That’s something we definitely improved upon,” Steinbrenner said “I didn’t spend $300 million to do it, but I think most people are pretty happy with Isiah and the job he’s done.”

As the Aug. 2 trade deadline approaches, New York may seek a corner outfielder. Joey Gallo began Wednesday with a .165 average, nine homers, 18 RBIs and 89 strikeouts in 194 at-bats.

“It’s been difficult for him, there’s no doubt about it. But there’s a lot of baseball to be had this season.” Steinbrenner said, “I still expect great things out of him. ... I am hesitant to give up top prospects. I always have been. But we’ve done it in the past. and all that will be on the table, too.”

Steinbrenner anticipates the path to a title will include a matchup against Houston, which beat the Yankees in the 2017 and 2019 AL Championship Series.

“We all need to be concerned about Houston. They’re a very good team,” he said. “But it’s going to be a challenge for them, too, make no mistake.”

He wouldn’t mind a Subway Series against the Mets, who entered Wednesday with the second-best record in the NL at 50-31.

“I’m all for it,” Steinbrenner said. “Means I’ll be there.”