BRIDGEWATER, N.J. — Gerrit Cole is on the clock and believes he’s “definitely close” to rejoining what surprisingly has been baseball’s best starting staff.

Recovering from a spring-training elbow injury, the reigning American League Cy Young winner saw his first game action of 2024 when he tossed 45 pitches (34 strikes) over 3 ¹/₃ scoreless innings — while topping out at 97 mph — to start a rehab assignment for Double-A Somerset against Hartford.

Cole allowed two singles, walked none and struck out five before leaving to a standing ovation at TD Bank Ballpark with one out in the fourth.

“I haven’t gone through this process in a while, so I’m taking it one start at a time and I’ll let my body tell me when we’re ready,” Cole said. “I had a lot of adrenaline and had to hold a few back. That’s a good place to be.

“Certainly, if you give me a number scale of where I was six weeks ago, I’d say [I’m] definitely close. I’m not sure how many more we’ll need, but definitely closer rather than farther away. It should be right around the corner now.”

Aaron Boone had said before the Yankees’ game against the Twins in The Bronx that he expects Cole to make “at least a few” minor league appearances, perhaps setting up the $324 million righty for a late June return.

Cole also stressed it would “not be a responsible move” to come back after just this one outing.

The first-place Yanks have up to 30 days of allowable rehab time for Cole, who went 15-4 with a 2.63 ERA and 222 strikeouts in an AL-high 207 innings in 2023.

The six-time All-Star was shut down in mid-March with nerve inflammation and edema in his pitching elbow and resumed playing catch on April 8 before throwing his first bullpen session in early May and facing batters for the first time on May 21.

He threw 43 pitches in a simulated game at the Yankees’ player development complex last Thursday and then a bullpen session Sunday, the final hurdle before heading to the minors.

The Yanks entered Tuesday with an MLB-best 2.78 staff ERA, but with Clarke Schmidt expected to be sidelined until August with a lat strain, Cody Poteet started Saturday against the Giants, and he likely will remain in that rotation spot until Cole is ready to return.

“That’s something we’ll work through as we go,” Boone said. “When do we want to take him? How many pitches do we want him built up to? Those will all be factored in and we’ll work with Gerrit through that, as well.”

Pitching to catching prospect Ben Rice, the 33-year-old Cole came out firing strikes, throwing just one ball among 10 pitches in the first inning.

The 12-year veteran allowed a leadoff single to Adael Amador before recording two Ks, reaching 97 mph on each of the strikeout pitches.

That was followed by a 1-2-3 second inning on 15 pitches, and 15 more in the third, including his second single allowed and his fourth strikeout.

Cole came out to face one more batter in the fourth, fanning Sterlin Thompson before Somerset manager Raul Dominguez replaced him with reliever Zach Messinger as the crowd stood and roared.

“I thought it was really fun overall. It was a nice environment, and it felt really great to be back out there and get back in the flow of things,” Cole said. “[I was happy with] everything. I threw all my pitches for strikes. … I was able to move the ball to both sides of the plate, got the bottom and top of the strike zone, so it was good.

“It was just a lively environment, a nice summer night, the big club is having a good night, so all is well.”