PHOENIX — Hideki Okajima is sound of mind and shoulder and ready to pitch for the A’s.

On Tuesday morning, he made the first step. He passed his physical, signed a minor league deal and dressed to take part in the first spring training session for A’s pitchers and catchers.

That must come as something of a surprise to the New York Yankees.

It was at about this time a year ago that the Yankees, who had invited former Boston Red Sox short reliever Okajima to spring training, sent him packing, saying that he’d failed his physical exam.

The Yankees found the left-hander’s shoulder ailing, and he headed back to Japan, wondering if his big league career was over.

“I wondered `Is this it for the Major Leagues?’ ” Okajima, talking through an interpreter, said Tuesday on arriving at Oakland’s Phoenix Municipal Stadium home.

He caught on with Softbank in Japan and put together a stellar year — a 0.96 ERA in 56 games and 47ï»¿2/3 innings pitching mostly situational relief. And he did it all with the shoulder the Yankees had deemed dicey.

“I was very surprised to be told I’d failed the physical,” Okajima said. “I was able to go to Softbank and have a good year, but I always wanted to return to the Major Leagues.”

“The shoulder is fine,” he said. “I pitched with no problem (in Japan). I won’t throw off a mound today, but I’m on a regular schedule.”

Manager Bob Melvin already was well-stocked with left-handed relievers in Jerry Blevins, Sean Doolittle, Pedro Figueroa, Travis Blackley and Jordan Norberto. But he’s enthused by the addition of Okajima, who is battle-tested in the post-season.

“I’m excited talking about the depth this move gives,” Melvin said. “His numbers in Japan were off the chart. He’s 37, but he doesn’t appear to have a 37-year-old’s body. And now our depth on the left-hand side is very strong.”

A’s outfielder Coco Crisp, who was with the Red Sox for Okajima’s first two seasons back in 2007 and 2008, is a big fan.

“I saw what he did then, and he was outstanding,” Crisp said. “You have to like his arm.”

Unless, apparently, you are the Yankees.